diff --git "a/79E4T4oBgHgl3EQfdAwp/content/tmp_files/load_file.txt" "b/79E4T4oBgHgl3EQfdAwp/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/79E4T4oBgHgl3EQfdAwp/content/tmp_files/load_file.txt" @@ -0,0 +1,839 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf,len=838 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='05087v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='DG] 12 Jan 2023 SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES SIMON BRENDLE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We prove a scalar curvature rigidity theorem for convex polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The proof uses the Fredholm theory for Dirac operators on manifolds with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' A variant of a theorem of Fefferman and Phong plays a central role in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Introduction Let Ω be a compact polytope in Rn with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We may write Ω = � α∈A{uα ≤ 0}, where A is a finite set and the uα are linear functions in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each α ∈ A, we denote by Nα ∈ Sn−1 the outward- pointing unit normal vector to the halfspace {uα ≤ 0} with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let g be a Riemannian metric which is defined on an open set containing Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each α ∈ A, we denote by να the outward-pointing unit normal vector to the halfspace {uα ≤ 0} with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We will make the following assumption: Matching Angle Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If x is point in ∂Ω and α1, α2 ∈ A satisfy uα1(x) = uα2(x) = 0, then ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Here, the inner product ⟨να1, να2⟩ is computed with respect to the metric g, and the inner product ⟨Nα1, Nα2⟩ is the standard inner product in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that n ≥ 3 is an odd integer, and Ω is a compact polytope in Rn with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let g be a Riemannian metric which is defined on an open set containing Ω and has nonnegative scalar curvature at each point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each α ∈ A, we assume that the mean curvature of the hypersurface {uα = 0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, we assume that the Matching Angle Hypothesis is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then the Ricci tensor of g vanishes at each point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1 also holds in the even-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This can be seen by considering the Cartesian product Ω × [0, 1] ⊂ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Scalar curvature comparison theorems for polytopes were first studied in seminal work of Gromov [6],[7],[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Li [9] has used minimal surface techniques to prove a scalar curvature comparison theorem for certain polytopes in The author was support by the National Science Foundation under grant DMS-2103573 and by the Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The author acknowledges the hospitality of T¨ubingen University, where part of this work was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 1 2 SIMON BRENDLE dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Wang, Xie, and Yu [10] have proposed a different approach to this problem which is based on the study of Dirac operators on manifolds with corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In this paper, we describe another approach to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' As in [10], we employ a spinor approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In contrast to [10], we work with boundary value problems for Dirac operators on smooth domains, which are well understood thanks to the work of B¨ar and Ballmann [1],[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In the following, we outline the main steps involved in the proof of The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We approximate a given convex polytope Ω by a one-parameter family of smooth convex domains Ωλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' On each domain Ωλ, we solve the Dirac equation for an m-tuple of spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) with a suitable local boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' To prove the existence of a solution satisfying that particular boundary condition, we use the Fredholm theory developed by B¨ar and Ballmann [1],[2] together with the homotopy invariance of the Fredholm index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Having constructed an m-tuple of harmonic spinors on Ωλ satisfying this boundary condition, we apply a Weitzenb¨ock formula, and integrate over Ωλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The resulting integral formula contains a term involving the scalar curvature, as well as a boundary term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Unfortunately, it is not clear if the boundary term has a favorable sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We are able to control the boundary integral by adapting a theorem due to Fefferman and Phong [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' A boundary value problem for the Dirac operator on a smooth domain Let m = 2[ n 2 ] denote the dimension of the space of spinors on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , En} denote the standard basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Throughout this section, we fix an orthonormal basis {¯s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , ¯sm} of the space of spinors on flat Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We define ωaαβ = ⟨Ea · ¯sα, ¯sβ⟩ for a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , n and α, β = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The matrices ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , ωn ∈ End(Cm) are skew-Hermitian, so that ωaαβ = −ωaβα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, ωaωb + ωbωa = −2δab id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In other words, m � β=1 (ωaαβ ωbβγ + ωbαβ ωaβγ) = −2δab δαγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We begin by stating a basic algebraic fact which will be needed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Assume that n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then there is no non-zero element of End(Cm) which anti-commutes with ωa ∈ End(Cm) for each a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We recall the definition of the spin representation in odd dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , En} denote the standard basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , [n 2 ], we define wk = E2k−1 − iE2k ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The spinor space is defined as the ex- terior algebra Λ∗W, where W = span{wk : k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , [n 2 ]} ⊂ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , [n 2 ]}, we define a linear map Pk ∈ End(Λ∗W) by Pk(wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∧ wjr) = wk ∧ wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∧ wjr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 3 Moreover, for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , [n 2 ]}, we define a linear map Qk ∈ End(Λ∗W) by Qk(wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∧ wjr) = 0, Qk(wk ∧ wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∧ wjr) = wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∧ wjr for k /∈ {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , jr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then PkPl + PlPk = QkQl + QlQk = 0 and PkQl + QlPk = δkl id for k, l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , [n 2 ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Finally, we define a linear map S ∈ End(Λ∗W) so that S(wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∧ wjr) = � wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∧ wjr if r is even −wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∧ wjr if r is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, PkS + SPk = 0, QkS + SQk = 0, and S2 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Consequently, there is a natural algebra homomorphism from the Clifford algebra ClC(n) to End(Λ∗W) which maps wk to i √ 2 Pk, ¯wk to i √ 2 Qk, and En to iS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' It is well known (see [5], Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='9) that span{Pk1 · · · PkrQl1 · · · Qls : r + s is even} = End(ΛevenW) ⊕ End(ΛoddW) and span{Pk1 · · · PkrQl1 · · · Qls : r + s is odd} = Hom(ΛevenW, ΛoddW) ⊕ Hom(ΛoddW, ΛevenW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We claim that there is no non-zero element of End(Λ∗W) which anti-commutes with Pk, Qk, S for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , [n 2 ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that L ∈ End(Λ∗W) is such an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since L anti-commutes with S, it follows that L ∈ Hom(ΛevenW, ΛoddW)⊕Hom(ΛoddW, ΛevenW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since L anti-commutes with Pk, Qk for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , [n 2 ]}, it follows that L anti-commutes with every element of Hom(ΛevenW, ΛoddW)⊕Hom(ΛoddW, ΛevenW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This implies that L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Assume that Ω is a domain in Rn with smooth boundary ∂Ω = Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let g be a Riemannian metric on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We denote by ν the outward-pointing unit normal vector field with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let ∇ denote the spin connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The Dirac operator is defined by Ds = n � i=1 ei · ∇eis, where {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , en} is a local orthonormal frame on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The boundary Dirac operator DΣ is given by DΣs = n−1 � i=1 ν · ei · ∇eis + 1 2 H s at each point on Σ, where {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , en−1} is a local orthonormal frame on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In the remainder of this section, we consider the Dirac operator act- ing on m-tuples of spinors with a suitable local boundary condition of 4 SIMON BRENDLE Lopatinsky-Shapiro type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' To formulate the boundary condition, we assume that N : Σ → Sn−1 is a given smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Consider an m-tuple of spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' At each point on Σ, we define (χs)α = − n � a=1 m � β=1 ⟨N, Ea⟩ ωaαβ ν · sβ and (Bs)α = n−1 � i=1 n � a=1 m � β=1 ⟨dN(ei), Ea⟩ ωaαβ ei · sβ, where {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , en−1} is a local orthonormal frame on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The map χ is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, χ2 is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) and t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , tm) are two m- tuples of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We compute (χ2s)α = n � a,b=1 m � β,γ=1 ⟨N, Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ ν · ν · sγ = − n � a,b=1 m � β,γ=1 ⟨N, Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ sγ = −1 2 n � a,b=1 m � β,γ=1 ⟨N, Ea⟩ ⟨N, Eb⟩ (ωaαβ ωbβγ + ωbαβ ωaβγ) sγ = n � a,b=1 m � γ=1 ⟨N, Ea⟩ ⟨N, Eb⟩ δab δαγ sγ = sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, m � α=1 ⟨(χs)α, tα⟩ = − n � a=1 m � α,β=1 ⟨N, Ea⟩ ωaαβ ⟨ν · sβ, tα⟩ = − n � a=1 m � α,β=1 ⟨N, Ea⟩ ωaβα ⟨sβ, ν · tα⟩ = m � β=1 ⟨sβ, (χt)β⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Assume that x ∈ Σ and ξ ∈ TxΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then the map (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) �→ (ν · ξ · s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , ν · ξ · sm) maps the eigenspace of χ with eigenvalue 1 to the eigenspace of χ with eigenvalue −1, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In particular, the two eigenspaces have the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each vector ξ ∈ TxΣ, the map (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) �→ (ν · ξ · s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , ν · ξ · sm) anti-commutes with χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' From this, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The map B is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, χ and B commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , en−1} be a local orthonormal frame on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then m � α=1 ⟨(Bs)α, tα⟩ = n−1 � i=1 n � a=1 m � α,β=1 ⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, tα⟩ = n−1 � i=1 n � a=1 m � α,β=1 ⟨dN(ei), Ea⟩ ωaβα ⟨sβ, ei · tα⟩ = m � β=1 ⟨sβ, (Bt)β⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This shows that B is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, (χBs)α − (Bχs)α = − n−1 � i=1 n � a,b=1 n � β,γ=1 ⟨N, Ea⟩ ⟨dN(ei), Eb⟩ ωaαβ ωbβγ ν · ei · sγ + n−1 � i=1 n � a,b=1 n � β,γ=1 ⟨dN(ei), Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ ei · ν · sγ = − n−1 � i=1 n � a,b=1 n � β,γ=1 ⟨N, Ea⟩ ⟨dN(ei), Eb⟩ (ωaαβ ωbβγ + ωbαβ ωaβγ) ν · ei · sγ = 2 n−1 � i=1 n � a,b=1 n � γ=1 ⟨N, Ea⟩ ⟨dN(ei), Eb⟩ δab δαγ ν · ei · sγ = 2 n−1 � i=1 ⟨N, dN(ei)⟩ ν · ei · sα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Thus, χ and B commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' At this point, we recall a definition from linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 6 SIMON BRENDLE Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let V and W be finite-dimensional vector spaces of the same dime, each of them equipped with an inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The trace norm of a linear map L : V → W is defined by ∥L∥tr = supQ tr(QL), where the supremum is taken over all linear isometries Q : W → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Equivalently, ∥L∥tr can be characterized as the sum of the singular values of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' It is easy to see from the definition that the trace norm satisfies the tri- angle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) is an m-tuple of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then ���� m � α=1 ⟨(Bs)α, sα⟩ ���� ≤ ∥dN∥tr � m � α=1 |sα|2 � at each point x ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Here, ∥dN∥tr denotes the trace norm of the differential dN : TxΣ → TN(x)Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The tangent space TxΣ is equipped with the restric- tion of the inner product g, and the tangent space TN(x)Sn−1 is equipped with the restriction of the standard inner product on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Fix a point x ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We can find an orthonormal basis {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , en−1} of TxΣ so that dN(ei) = λi ˆEi, where { ˆE1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , ˆEn−1} is an orthonormal ba- sis of TN(x)Sn−1 and λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , λn−1 ≥ 0 denote the singular values of dN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then m � α=1 ���� n � a=1 m � β=1 ⟨ ˆEi, Ea⟩ ωaαβ ei · sβ ���� 2 = n � a,b=1 m � α,β,γ=1 ⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ ωaαβ ωbαγ ⟨ei · sβ, ei · sγ⟩ = − n � a,b=1 m � α,β,γ=1 ⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ ωaαβ ωbγα ⟨sβ, sγ⟩ = −1 2 n � a,b=1 m � α,β,γ=1 ⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ (ωaγα ωbαβ + ωbγα ωaαβ) ⟨sβ, sγ⟩ = n � a,b=1 m � β,γ=1 ⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ δab δγβ ⟨sβ, sγ⟩ = m � α=1 |sα|2 SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 7 for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using the Cauchy-Schwarz inequality, we obtain ���� n � a=1 m � α,β=1 ⟨ ˆEi, Ea⟩ ωaαβ ⟨ei · sβ, sα⟩ ���� ≤ � m � α=1 ���� n � a=1 m � β=1 ⟨ ˆEi, Ea⟩ ωaαβ ei · sβ ���� 2� 1 2 � m � α=1 |sα|2 � 1 2 = m � α=1 |sα|2 for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Summation over i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , n − 1 gives ���� m � α=1 ⟨(Bs)α, sα⟩ ���� = ���� n−1 � i=1 n � a=1 m � α,β=1 ⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, sα⟩ ���� = ���� n−1 � i=1 λi � n � a=1 m � α,β=1 ⟨ ˆEi, Ea⟩ ωaαβ ⟨ei · sβ, sα⟩ ����� ≤ � n−1 � i=1 λi � � m � α=1 |sα|2 � , as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) and t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , tm) are m-tuples of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then 0 = � Σ m � α=1 ⟨DΣsα, (χt)α⟩ dσg + � Σ m � α=1 ⟨(χs)α, DΣtα⟩ dσg + � Σ m � α=1 ⟨(Bs)α, tα⟩ dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Equivalently, 0 = � Σ m � α=1 ⟨(As)α, (χt)α⟩ dσg + � Σ m � α=1 ⟨(χs)α, (At)α⟩ dσg, where A is defined by A = DΣ + 1 2χB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , en−1} be a local orthonormal frame on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We define a tangential vector field Z on Σ by ⟨Z, ei⟩ = n � a=1 m � α,β=1 ⟨N, Ea⟩ ωaαβ ⟨ei · sβ, tα⟩ 8 SIMON BRENDLE for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then divΣZ = n−1 � i=1 n � a=1 m � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='β=1 ⟨N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Ea⟩ ωaαβ ⟨ei · ∇eisβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' tα⟩ + n−1 � i=1 n � a=1 m � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='β=1 ⟨N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Ea⟩ ωaαβ ⟨ei · sβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∇eitα⟩ − n � a=1 m � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='β=1 H ⟨N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Ea⟩ ωaαβ ⟨ν · sβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' tα⟩ + n−1 � i=1 n � a=1 m � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='β=1 ⟨dN(ei),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Ea⟩ ωaαβ ⟨ei · sβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' tα⟩ = − n−1 � i=1 m � β=1 ⟨ei · ∇eisβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ν · (χt)β⟩ + n−1 � i=1 m � α=1 ⟨ei · ν · (χs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' ∇eitα⟩ + m � α=1 H ⟨(χs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' tα⟩ + m � α=1 ⟨(Bs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' tα⟩ = m � β=1 ⟨DΣsβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' (χt)β⟩ + m � α=1 ⟨(χs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' DΣtα⟩ + m � α=1 ⟨(Bs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' tα⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Integrating over Σ, we obtain 0 = � Σ m � β=1 ⟨DΣsβ, (χt)β⟩ dσg + � Σ m � α=1 ⟨(χs)α, DΣtα⟩ dσg + � Σ m � α=1 ⟨(Bs)α, tα⟩ dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' It is well known that the boundary Dirac operator DΣ is formally self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='3 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='5 that χB is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Consequently, the operator A = DΣ+ 1 2χB is formally self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Finally, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='8 implies that A and χ anti-commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 9 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) is an m-tuple of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then − � Ω m � α=1 |Dsα|2 dvolg + � Ω n � α=1 |∇sα|2 dvolg + 1 4 � Ω m � α=1 R |sα|2 dvolg ≤ 1 2 � Σ m � α=1 ⟨DΣsα, sα − (χs)α⟩ dσg + 1 2 � Σ m � α=1 ⟨sα − (χs)α, DΣsα⟩ dσg − 1 2 � Σ (H − ∥dN∥tr) � m � α=1 |sα|2 � dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By the Weitzenb¨ock formula, D2sα = −∆sα + 1 4 R sα, where ∆ denotes the connection Laplacian on the spinor bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using the divergence theorem, we obtain − � Ω m � α=1 |Dsα|2 dvolg + � Ω m � α=1 |∇sα|2 dvolg + 1 4 � Ω m � α=1 R |sα|2 dvolg = � Σ m � α=1 ⟨ν · Dsα, sα⟩ dσg + � Σ m � α=1 ⟨∇νsα, sα⟩ dσg = � Σ ⟨DΣsα, sα⟩ dσg − 1 2 � Σ m � α=1 H |sα|2 dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='8 with s = t gives 0 = � Σ m � α=1 ⟨DΣsα, (χs)α⟩ dσg + � Σ m � α=1 ⟨(χs)α, DΣsα⟩ dσg + � Σ m � α=1 ⟨(Bs)α, sα⟩ dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This gives − � Ω m � α=1 |Dsα|2 dvolg + � Ω n � α=1 |∇sα|2 dvolg + 1 4 � Ω m � α=1 R |sα|2 dvolg = 1 2 � Σ m � α=1 ⟨DΣsα, sα⟩ dσg + 1 2 � Σ m � α=1 ⟨sα, DΣsα⟩ dσg − 1 2 � Σ m � α=1 H |sα|2 dσg = 1 2 � Σ m � α=1 ⟨DΣsα, sα − (χs)α⟩ dσg + 1 2 � Σ m � α=1 ⟨sα − (χs)α, DΣsα⟩ dσg − 1 2 � Σ m � α=1 ⟨(Bs)α, sα⟩ dσg − 1 2 � Σ m � α=1 H |sα|2 dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Hence, the assertion follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 10 SIMON BRENDLE Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that R ≥ 0 at each point in Ω and H ≥ ∥dN∥tr at each point on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then every m-tuple of harmonic spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) with χs = s is parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Replacing N by −N, we can draw the following conclusion: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that R ≥ 0 at each point in Ω and H ≥ ∥dN∥tr at each point on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then every m-tuple of harmonic spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) with χs = −s is parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that Ω is a convex domain in Rn with smooth boundary ∂Ω = Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let g be a Riemannian metric on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that N : Σ → Sn−1 is a smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then the boundary condition χs = s is a D-elliptic boundary condition in the sense of B¨ar and Ballmann [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We apply Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='18 in [2] with E′ = ker(id − χ) and E′′ = ker(id + χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='4 implies that, for each point x ∈ Σ and each ξ ∈ TxΣ, the map (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) �→ (ν · ξ · s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , ν · ξ · sm) interchanges ker(id − χ) and ker(id + χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Therefore, the boundary condition χs = s is a D-elliptic boundary condition in the sense of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Assume that n ≥ 3 is an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that Ω is a convex domain in Rn with smooth boundary ∂Ω = Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let g be a Riemannian metric on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that N : Σ → Sn−1 is homotopic to the Gauss map of Σ with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then the Dirac operator with the boundary condition χs = s has Fredholm index at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since the Fredholm index is homotopy invariant, it suffices to prove the assertion in the special case when g is the Euclidean metric and N is the Gauss map of Σ with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We first analyze the kernel of the Dirac operator with the boundary con- dition χs = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Recall that ¯s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , ¯sm is a basis of spinors on flat Rn, and ωaαβ = ⟨Ea · ¯sα, ¯sβ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, ¯s = (¯s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , ¯sm) is an m-tuple of harmonic spinors on Ω which satisfies the boundary condition χ¯s = ¯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Therefore, the kernel has dimension at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We next examine the cokernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The cokernel can be identified with the space of all m-tuples of harmonic spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) such that ⟨ν · s, t⟩ = 0 for all points x ∈ Σ and all t ∈ ker(id − χ) (see [2], Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We claim that this space has dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' To see this, suppose that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) is an m-tuple of harmonic spinors such that ⟨ν · s, t⟩ = 0 for all points x ∈ Σ and all t ∈ ker(id − χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This implies s ∈ ker(id + χ) at each point on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since H = ∥dN∥tr at each point on Σ, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='12 im- plies that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm) is parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In other words, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , sm are constant spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us write sα = �m β=1 zαβ ¯sβ for some matrix z ∈ End(Cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since χs = −s at each point on Σ, it follows that the matrix z ∈ End(Cm) anti- commutes with the matrix �n a=1⟨N(x), Ea⟩ ωa ∈ End(Cm) for each point x ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' It is easy to see that the Gauss map N : Σ → Sn−1 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 11 Consequently, the matrix z ∈ End(Cm) anti-commutes with ωa ∈ End(Cm) for each a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since n is odd, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1 implies that z = 0, hence s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This shows that the cokernel has dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Approximating a compact, convex polytope by smooth domains Let us consider a compact, convex polytope Ω ⊂ Rn with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We write Ω = � α∈A{uα ≤ 0}, where A is a finite set and the uα are linear functions in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' After eliminating redundant inequalities, we may assume that the following condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each α ∈ A, the set Ω ∩ {uα > 0} is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let g be a Riemannian metric which is defined on an open set containing Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each α ∈ A, ∇uα will denote the gradient of uα with respect to the metric g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' |∇uα| will denote the norm of the gradient of uα with respect to the metric g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' and να = ∇uα |∇uα| will denote the outward-pointing unit normal vector to the halfspace {uα ≤ 0} with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each α ∈ A, we denote by Nα ∈ Sn−1 the outward-pointing unit normal vector to the halfspace {uα ≤ 0} with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each λ > 0, the function � α∈A eλuα is convex with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, � α∈A eλuα > 1 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, we can find large number λ0 such that infΩ � α∈A eλuα < 1 for each λ > λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each λ > λ0, we define Ωλ = � � α∈A eλuα ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each λ > λ0, Ωλ is a convex domain in Rn with smooth boundary Σλ = ∂Ωλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The sets Ωλ form an increasing family of sets in the sense that Ωλ ⊂ Ωµ for λ0 < λ < µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, � λ>λ0 Ωλ = � α∈A {uα < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If λ is sufficiently large, then infΣλ �� � α∈A eλuα duα �� ≥ C−1 for some large constant C which is independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that the assertion is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then there exists a sequence of positive real numbers λl → ∞ and a se- quence of points xl ∈ Σλl such that �� � α∈A eλuα duα �� ≤ l−1 at the point xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' After passing to a subsequence, we may assume that the sequence xl converges to a point x0 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, we may assume that, for each α ∈ A, the sequence eλluα(xl) converges to a nonnegative real number zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since � α∈A eλluα(xl) = 1 for each l, we know that � α∈A zα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let A0 := {α ∈ A : zα > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, A0 is non-empty, and uα(x0) = 0 for all 12 SIMON BRENDLE α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, � α∈A0 zα duα = 0 at the point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' On the other hand, since Ω is a convex set with non-empty interior, we can find a tangent vector ξ ∈ Tx0Ω such that duα(ξ) > 0 for all α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If λ is sufficiently large, then infΣλ �� � α∈A eλuα |∇uα| Nα �� ≥ C−1 for some large constant C which is independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that the assertion is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then there exists a sequence of positive real numbers λl → ∞ and a se- quence of points xl ∈ Σλl such that �� � α∈A eλuα |∇uα| Nα �� ≤ l−1 at the point xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' After passing to a subsequence, we may assume that the sequence xl converges to a point x0 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, we may assume that, for each α ∈ A, the sequence eλluα(xl) |∇uα(xl)| converges to a nonnegative real num- ber zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since � α∈A eλluα(xl) = 1 for each l, we know that � α∈A zα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let A0 := {α ∈ A : zα > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, A0 is non-empty, and uα(x0) = 0 for all α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, � α∈A0 zαNα = 0 at the point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' On the other hand, since Ω is a convex set with non-empty interior, we can find a vector ξ ∈ Rn such that ⟨Nα, ξ⟩ > 0 for all α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The outward-pointing unit normal vector to the domain Ωλ with respect to the metric g is given by ν = � α∈A eλuα ∇uα �� � α∈A eλuα ∇uα �� = � α∈A eλuα |∇uα| να �� � α∈A eλuα |∇uα| να ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We define a map N : Σλ → Sn−1 by N = � α∈A eλuα |∇uα| Nα �� � α���A eλuα |∇uα| Nα ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The map N : Σλ → Sn−1 is homotopic to the Gauss map of Σλ with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In the special case when g is the Euclidean metric, the map N coincides with the Gauss map of Σλ, and the assertion is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' To prove the assertion in general, we deform the metric g to the Euclidean met- ric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let x ∈ Σλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let π : TxΩ → TxΩ denotes the orthogonal projection to the orthogonal complement of ν and P : Rn → Rn denotes the orthogonal projection to the orthogonal complement of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then H − SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 13 ∥dN∥tr ≥ Vλ, where the function Vλ : Σλ → R is defined by Vλ = λ � α∈A eλuα |∇uα|2 |π(να)|2 �� � α∈A eλuα |∇uα| να �� − λ � α∈A eλuα |∇uα|2 |π(να)| |P(Nα)| �� � α∈A eλuα |∇uα| Nα �� + � α∈A eλuα (∆uα − (D2uα)(ν, ν)) �� � α∈A eλuα |∇uα| να �� − � α∈A eλuα |∇(|∇uα|)| |P(Nα)| �� � α∈A eλuα |∇uα| Nα �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , en−1} denote a local orthonormal frame on Σλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The mean curvature of Σλ is given by H = λ �n−1 i=1 � α∈A eλuα ⟨∇uα, ei⟩2 �� � α∈A eλuα ∇uα �� + �n−1 i=1 � α∈A eλuα (D2uα)(ei, ei) �� � α∈A eλuα ∇uα �� = λ � α∈A eλuα |π(∇uα)|2 �� � α∈A eλuα ∇uα �� + � α∈A eλuα (∆uα − (D2uα)(ν, ν)) �� � α∈A eλuα ∇uα �� = λ � α∈A eλuα |∇uα|2 |π(να)|2 �� � α∈A eλuα |∇uα| να �� + � α∈A eλuα (∆uα − (D2uα)(ν, ν)) �� � α∈A eλuα |∇uα| να �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If ξ is a tangent vector to Σλ, then dN(ξ) = λ � α∈A eλuα |∇uα| ⟨∇uα, ξ⟩ P(Nα) �� � α∈A eλuα |∇uα| Nα �� + � α∈A eλuα ⟨∇(|∇uα|), ξ⟩ P(Nα) �� � α∈A eλuα |∇uα| Nα �� = λ � α∈A eλuα |∇uα|2 ⟨π(να), ξ⟩ P(Nα) �� � α∈A eλuα |∇uα| Nα �� + � α∈A eλuα ⟨∇(|∇uα|), ξ⟩ P(Nα) �� � α∈A eλuα |∇uα| Nα �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The trace norm of a linear transformation of the form ξ �→ ⟨X, ξ⟩ Y is given by |X| |Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since the trace norm satisfies the triangle inequality, it follows that ∥dN∥tr ≤ λ � α∈A eλuα |∇uα|2 |π(να)| |P(Nα)| �� � α∈A eλuα |∇uα| Nα �� + � α∈A eλuα |∇(|∇uα|)| |P(Nα)| �� � α∈A eλuα |∇uα| Nα �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Putting these facts together, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In the following, we denote by Vλ,− = max{−Vλ, 0} the negative part of Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that the Matching Angle Hypothesis is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then supΣλ Vλ,− ≤ o(λ) as λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that the assertion is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then there exists a sequence of positive real numbers λl → ∞ and a se- quence of points xl ∈ Σλl such that lim supl→∞ λ−1 l Vλl(xl) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' After passing to a subsequence, we may assume that the sequence xl converges to a point x0 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, we may assume that, for each α ∈ A, the sequence eλluα(xl) |∇uα(xl)| converges to a nonnegative real number zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 14 SIMON BRENDLE Since � α∈A eλluα(xl) = 1 for each l, we know that � α∈A zα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let A0 := {α ∈ A : zα > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, A0 is non-empty, and uα(x0) = 0 for all α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The Matching Angle Hypothesis implies that, at the point x0, ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ for all α1, α2 ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let π : Tx0Ω → Tx0Ω denote the orthogonal projection to the orthogonal complement of � α∈A0 zανα, and let P : Rn → Rn denote the orthogonal projection to the orthogonal complement of � α∈A0 zαNα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each β ∈ A0, we have |π(νβ)|2 = 1 − � � α∈A0 zανα, νβ �2 �� � α∈A0 zανα ��2 = 1 − � � α∈A0 zαNα, Nβ �2 �� � α∈A0 zαNα ��2 = |P(Nβ)|2 at the point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Therefore, for each β ∈ A0, we obtain |π(νβ)| �� � α∈A0 zανα �� = |P(Nβ)| �� � α∈A0 zαNα �� at the point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='5, we conclude that λ−1 l Vλl(xl) → 0 as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In the remainder of this section, we will estimate the Ls-norm Vλ,− on Σλ∩ Br(p), where s ∈ [1, 3 2) is a fixed exponent and Br(p) denotes a Euclidean ball of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We begin by recalling a basic fact about the area of convex hypersurfaces in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let Br(p) denote a Euclidean ball of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then the in- tersection Σλ ∩ Br(p) has area at most Crn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This follows from the fact that the hypersurface Σλ = ∂Ωλ is outward-minimizing with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Consider three pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We denote by G(α1,α2,α3) λ the set of all points x ∈ Σλ with the property that uα1(x) ≥ uα2(x) ≥ uα3(x) and uα3(x) ≥ uα(x) for α ∈ A \\ {α1, α2, α3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Assume that the mean curvature of the hypersurface {uα = 0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us fix an exponent s ∈ [1, 3 2), and let Br(p) denote a Euclidean ball of radius r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If λr is sufficiently large, then � rs+1−n � G(α1,α2,α3) λ ∩{uα2≤−λ− 7 8 r 1 8 }∩Br(p) V s λ,− � 1 s ≤ Cλr e−(λr) 1 8 for all pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us consider an arbitrary point x ∈ G(α1,α2,α3) λ with uα2(x) ≤ −λ− 7 8 r 1 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By definition of G(α1,α2,α3) λ , it follows that uα(x) ≤ −λ− 7 8r 1 8 for all α ∈ A \\ {α1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using the identity � α∈A eλuα(x) = 1, we obtain uα1(x) ≥ −Cλ−1 e−(λr) 1 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, |ν − να1| ≤ C e−(λr) 1 8 and |N − Nα1| ≤ SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 15 C e−(λr) 1 8 at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' From this, we deduce that |π(να1)| ≤ C e−(λr) 1 8 and |P(Nα1)| ≤ C e−(λr) 1 8 at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Therefore, Vλ ≥ ∆uα1 − (D2uα1)(να1, να1) |∇uα1| − Cλ e−(λr) 1 8 at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since uα1(x) ≥ −Cλ−1 e−(λr) 1 8 and uα(x) ≤ −λ− 7 8r 1 8 for all α ∈ A \\ {α1}, we can find a point y ∈ Ω such that uα1(y) = 0 and d(x, y) ≤ Cλ−1 e−(λr) 1 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By assumption, the mean curvature of the hyper- surface {uα1 = 0} at the point y is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This implies ∆uα1 − (D2uα1)(να1, να1) |∇uα1| ≥ 0 at the point y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Consequently, ∆uα1 − (D2uα1)(να1, να1) |∇uα1| ≥ −C d(x, y) at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Thus, we conclude that Vλ(x) ≥ −Cλ e−(λr) 1 8 for each point x ∈ G(α1,α2,α3) λ ∩ {uα2 ≤ −λ− 7 8r 1 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' On the other hand, Σλ ∩ Br(p) has area at most Crn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Consequently, � rs+1−n � G(α1,α2,α3) λ ∩{uα2≤−λ− 7 8 r 1 8 }∩Br(p) V s λ,− � 1 s ≤ Cλr e−(λr) 1 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Assume that the Matching Angle Hypothesis holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us fix an exponent s ∈ [1, 3 2), and let Br(p) denote a Euclidean ball of radius r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If λr is sufficiently large, then � rs+1−n � G(α1,α2,α3) λ ∩{uα2≥−λ− 7 8 r 1 8 }∩{uα3≤−λ− 3 4 r 1 4 }∩Br(p) V s λ,− � 1 s ≤ C (λr) 1 8 − 7 8s for all pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We distinguish two cases: Case 1: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By continuity, we can find a real number δ such that Ω ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If λr is sufficiently large, then λ− 7 8r 1 8 ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This implies G(α1,α2,α3) λ ∩ {uα2 ≥ −λ− 7 8 r 1 8 } ⊂ Σλ ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Hence, the assertion is trivially true in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 16 SIMON BRENDLE Case 2: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' It follows from Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1 that the hypersurfaces {uα1 = 0} and {uα2 = 0} intersect transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us consider an arbitrary point x ∈ G(α1,α2,��3) λ with uα2(x) ≥ −λ− 7 8r 1 8 and uα3(x) ≤ −λ− 3 4r 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, uα1(x) ≥ −λ− 7 8r 1 8 by definition of G(α1,α2,α3) λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, uα(x) ≤ −λ− 3 4r 1 4 for all α ∈ A \\ {α1, α2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Consequently, we can find a point y ∈ Ω such that uα1(y) = uα2(y) = 0 and d(x, y) ≤ Cλ− 7 8r 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The Matching Angle Hypothesis implies ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ at the point y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Consequently, |⟨να1, να2⟩ − ⟨Nα1, Nα2⟩| ≤ C d(x, y) at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' From this, we deduce that |π(να1)| �� � α∈A eλuα |∇uα| να �� − |P(Nα1)| �� � α∈A eλuα |∇uα| Nα �� ≥ −C d(x, y) − C e−(λr) 1 4 and |π(να2)| �� � α∈A eλuα |∇uα| να �� − |P(Nα2)| �� � α∈A eλuα |∇uα| Nα �� ≥ −C d(x, y) − C e−(λr) 1 4 at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Thus, we conclude that Vλ(x) ≥ −Cλ 1 8 r− 7 8 for each point x ∈ G(α1,α2,α3) λ ∩ {uα2 ≥ −λ− 7 8 r 1 8 } ∩ {uα3 ≤ −λ− 3 4r 1 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By transversality, the set {0 ≥ uα1 ≥ −λ− 7 8r 1 8} ∩ {0 ≥ uα2 ≥ −λ− 7 8 r 1 8} ∩ Br(p) can be covered by C (λr) 7(n−2) 8 Euclidean balls of radius λ− 7 8 r 1 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' More- over, the intersection of Σλ with each ball of radius λ− 7 8r 1 8 has area at most C (λr)− 7(n−1) 8 rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This shows that Σλ ∩ {uα1 ≥ −λ− 7 8 r 1 8 } ∩ {uα2 ≥ −λ− 7 8 r 1 8 } ∩ Br(p) has area at most C (λr)− 7 8 rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since G(α1,α2,α3) λ ∩ {uα2 ≥ −λ− 7 8r 1 8} ∩ Br(p) ⊂ Σλ ∩ {uα1 ≥ −λ− 7 8 r 1 8 } ∩ {uα2 ≥ −λ− 7 8r 1 8} ∩ Br(p), it follows that � rs+1−n � G(α1,α2,α3) λ ∩{uα2≥−λ− 7 8 r 1 8 }∩{uα3≤−λ− 3 4 r 1 4 }∩Br(p) V s λ,− � 1 s ≤ C (λr) 1 8 − 7 8s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us fix an exponent s ∈ [1, 3 2), and let Br(p) denote a Euclidean ball of radius r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If λr is sufficiently large, then � rs+1−n � G(α1,α2,α3) λ ∩{uα3≥−λ− 3 4 r 1 4 }∩Br(p) V s λ,− � 1 s ≤ C (λr)1− 3 2s for all pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We distinguish two cases: Case 1: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ∩ {uα3 = 0} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By continuity, we can find a real number δ such that Ω ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} ∩ {uα3 ≥ −δ} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If λr is sufficiently large, then λ− 3 4r 1 4 ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This implies G(α1,α2,α3) λ ∩ {uα3 ≥ −λ− 3 4 r 1 4 } ⊂ Σλ ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} ∩ {uα3 ≥ −δ} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Hence, the assertion is trivially true in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Case 2: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ∩ {uα3 = 0} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' It follows from Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1 that the hypersurfaces {uα1 = 0}, {uα2 = 0}, and {uα3 = 0} intersect transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us consider an arbitrary point x ∈ G(α1,α2,α3) λ with uα3(x) ≥ −λ− 3 4r 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, Vλ(x) ≥ −Cλ for all points x ∈ G(α1,α2,α3) λ ∩ {uα3 ≤ −λ− 3 4 r 1 4 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By transversality, the set {0 ≥ uα1 ≥ −λ− 3 4r 1 4}∩{0 ≥ uα2 ≥ −λ− 3 4 r 1 4}∩{0 ≥ uα3 ≥ −λ− 3 4 r 1 4 }∩Br(p) can be covered by C (λr) 3(n−3) 4 Euclidean balls of radius λ− 3 4 r 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' More- over, the intersection of Σλ with each ball of radius λ− 3 4r 1 4 has area at most C (λr)− 3(n−1) 4 rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This shows that Σλ ∩ {uα1 ≥ −λ− 3 4 r 1 4 } ∩ {uα2 ≥ −λ− 3 4 r 1 4 }∩{uα3 ≥ −λ− 3 4 r 1 4 }∩Br(p) has area at most C (λr)− 3 2 rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since G(α1,α2,α3) λ ∩ {uα3 ≥ −λ− 3 4 r 1 4 } ∩ Br(p) ⊂ Σλ ∩ {uα1 ≥ −λ− 3 4 r 1 4 } ∩ {uα2 ≥ −λ− 3 4 r 1 4 } ∩ {uα3 ≥ −λ− 3 4r 1 4} ∩ Br(p), it follows that � rs+1−n � G(α1,α2,α3) λ ∩{uα3≥−λ− 3 4 r 1 4 }∩Br(p) V s λ,− � 1 s ≤ C (λr)1− 3 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Assume that the mean curvature of the hypersurface {uα = 0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0} and that the Matching Angle Hypothesis is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us fix an exponent s ∈ [1, 3 2), and let Br(p) denote a Euclidean ball of radius r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If λr is sufficiently large, then � rs+1−n � Σλ∩Br(p) V s λ,− � 1 s ≤ C (λr)−1 + C (λr) 1 8 − 7 8s + C (λr)1− 3 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 18 SIMON BRENDLE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='9, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='10, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='11, we con- clude that � rs+1−n � G(α1,α2,α3) λ ∩Br(p) V s λ,− � 1 s ≤ C (λr)−1 + C (λr) 1 8− 7 8s + C (λr)1− 3 2s for all pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' On the other hand, Σλ = � α1,α2,α3 G(α1,α2,α3) λ , where the union is taken over all pairwise distinct el- ements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Hence, the assertion follows by summation over α1, α2, α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Assume that the mean curvature of the hypersurface {uα = 0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0} and that the Matching Angle Hypothesis is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us fix an exponent s ∈ [1, 3 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then sup p∈Rn sup r≤1 � rs+1−n � Σλ∩Br(p) V s λ,− � 1 s → 0 as λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us consider an arbitrary sequence λl → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='6, we can find a sequence of positive real numbers δl → 0 such that sup p∈Rn sup r≤(δlλl)−1 � rs+1−n � Σλl∩Br(p) V s λl,− � 1 s → 0 as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' On the other hand, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='12 implies that sup p∈Rn sup (δlλl)−1≤r≤1 � rs+1−n � Σλl∩Br(p) V s λl,− � 1 s → 0 as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Putting these facts together, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1 Throughout this section, we assume that n ≥ 3 is an odd integer, and Ω is a compact polytope in Rn with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let g be a Riemannian metric which is defined on an open set containing Ω and has nonnegative scalar curvature at each point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We assume that the mean curvature of the hypersurface {uα = 0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0} and that the Matching Angle Hypothesis is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let U denote a Euclidean ball such that the closure of U is contained in the interior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Consider a sequence λl → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Note that U ⊂ Ωλl if l is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='14 we can find an m-tuple of harmonic spinors s(l) = (s(l) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , s(l) m ) such that s(l) is defined on Ωλl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' s(l) does not vanish identically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Ds(l) = 0 in Ωλl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' and χs(l) = s(l) on Σλl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Standard SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 19 unique continuation arguments imply that � U �m α=1 |s(l) α |2 dvolg > 0 if l is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' By scaling, we can arrange that � U m � α=1 |s(l) α |2 dvolg = 1 for each l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='10, we obtain � Ωλl m � l=1 |∇s(l) α |2 dvolg + 1 4 � Ωλl m � α=1 R |s(l) α |2 dvolg ≤ −1 2 � Σλl (H − ∥dN∥tr) � m � α=1 |s(l) α |2 � dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='5 implies that H − ∥dN∥tr ≥ Vλl at each point on Σλl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Con- sequently, � Ωλl m � l=1 |∇s(l) α |2 dvolg + 1 4 � Ωλl m � α=1 R |s(l) α |2 dvolg ≤ 1 2 � Σλl Vλl,− � m � α=1 |s(l) α |2 � dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Note that the hypersurface Σλl = ∂Ωλl can be written as a radial graph with bounded slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' From this, it is easy to see that Ωλl is bi-Lipschitz equivalent to the Euclidean unit ball, with constants that are independent of λl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='7 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='13, we obtain � Σλl Vλl,− F 2 dσg ≤ εl � Ωλl |∇F|2 dvolg + εl � � Σλl F dσg �2 for every smooth function F on Ωλl, where εl → 0 as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, the Sobolev trace theorem implies � � Σλl F dσg �2 ≤ C � Ωl |∇F|2 dvolg + C � U F 2 dvolg for every smooth function F on Ωλl, where C is a uniform constant inde- pendent of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Putting these facts together, we conclude that � Σλl Vλl,− F 2 dσg ≤ Cεl � Ωλl |∇F|2 dvolg + Cεl � U F 2 dvolg 20 SIMON BRENDLE for every smooth function F on Ωλl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In the next step, we apply this inequal- ity with F = � δ2 + �m α=1 |s(l) α |2� 1 2 , and send δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This gives � Ωλl m � l=1 |∇s(l) α |2 dvolg + 1 4 � Ωλl m � α=1 R |s(l) α |2 dvolg ≤ 1 2 � Σλl Vλl,− � m � α=1 |s(l) α |2 � dσg ≤ Cεl � Ωλl m � α=1 |∇s(l) α |2 dvolg + Cεl � U m � α=1 |s(l) α |2 dvolg for each l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Since the scalar curvature is nonnegative, it follows that � Ωλl m � α=1 |∇s(l) α |2 dvolg ≤ Cεl � U m � α=1 |s(l) α |2 dvolg if l is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Passing to the limit as l → ∞, we obtain a non- vanishing m-tuple of parallel spinors defined on the interior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Conse- quently, the Ricci tensor of g vanishes at each point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' A variant of a theorem of Fefferman and Phong In this section, we describe a variant of an estimate due to Fefferman and Phong [4], which plays a central role in our argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We denote by Q the collection of all (n − 1)-dimensional cubes of the form [2mj1, 2m(j1 + 1)] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' × [2mjn−1, 2m(jn−1 + 1)] × {0}, where m ∈ Z and j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , jn−1 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each Q ∈ Q, we denote by |Q| the (n − 1)-dimensional volume of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Fix an exponent s ∈ (1, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let V be a nonnegative continuous function on the hyperplane Rn−1 × {0} ⊂ Rn with the property that diam(Q)s+1−n � Q V s ≤ 1 for each (n − 1)-dimensional cube Q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Suppose that F is a smooth function on the half-space Rn + = {x ∈ Rn : xn ≥ 0}, and let f denote the restriction of F to the boundary ∂Rn + = Rn−1 × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Then � Q V f 2 ≤ C � Q×[0,diam(Q)] |∇F|2 + C diam(Q)−1 |Q|−1 � � Q |f| �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' for each (n − 1)-dimensional cube Q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 21 The proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='1 involves a straightforward adaptation of the arguments of Fefferman and Phong [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let us fix an exponent t such that s > t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We define a nonnegative function W : Rn−1 × {0} → R by W(x) = sup Q∈Q,x∈Q � |Q|−1 � Q V s � 1 s for each point x ∈ Rn−1 × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In other words, W s is the maximal function associated with the function V s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, V ≤ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We assume that F is a smooth function on the half-space Rn + = {x ∈ Rn : xn ≥ 0}, and let f denote the restriction of F to the boundary ∂Rn + = Rn−1 × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each (n − 1)-dimensional cube Q ∈ Q, we denote by fQ = |Q|−1 � Q f the mean value of f over the cube Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each (n − 1)-dimensional cube Q0 ∈ Q, we have � |Q0|−1 � Q0 W t � 1 t ≤ C sup Q∈Q,Q0⊂Q � |Q|−1 � Q V s � 1 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For abbreviation, let Λ = sup Q∈Q,Q0⊂Q � |Q|−1 � Q V s � 1 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We define a nonnegative function W0 : Q0 → R by W0(x) = sup Q∈Q,x∈Q⊂Q0 � |Q|−1 � Q V s � 1 s for each point x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Clearly, W(x) = max{Λ, W0(x)} for each point x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' The Hardy-Littlewood maximal inequality implies |Q0|−1 |{x ∈ Q0 : W0(x)s > α}| ≤ Cα−1 |Q0|−1 � Q0 V s ≤ Cα−1 Λs for all α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We multiply both sides by α t s−1 and integrate over α ∈ [Λs, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This gives |Q0|−1 � Q0 W t 0 ≤ C Λt, hence |Q0|−1 � Q0 W t ≤ C Λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 22 SIMON BRENDLE Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Given a real number ε > 0, we can find a real number δ > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If Q0 is an (n − 1)-dimensional cube in Q and A ⊂ Q0 is a Borel set with |A| ≤ δ |Q0|, then � A W ≤ ε � Q0 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='2, we obtain � |Q0|−1 � Q0 W t � 1 t ≤ C sup Q∈Q,Q0⊂Q � |Q|−1 � Q V s � 1 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, sup Q∈Q,Q0⊂Q � |Q|−1 � Q V s � 1 s ≤ inf Q0 W ≤ |Q0|−1 � Q0 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Therefore, � |Q0|−1 � Q0 W t � 1 t ≤ C |Q0|−1 � Q0 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Hence, if A ⊂ Q0 is a Borel set with |A| ≤ δ Q0, then � A W ≤ |A| t−1 t � � Q0 W t � 1 t ≤ δ t−1 t |Q0| t−1 t � � Q0 W t � 1 t ≤ Cδ t−1 t � Q0 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each (n − 1)-dimensional cube Q0 ∈ Q, we have |Q0|−1 � Q0 W ≤ C diam(Q0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='2, we obtain � |Q0|−1 � Q0 W t � 1 t ≤ C sup Q∈Q,Q0⊂Q � |Q|−1 � Q V s � 1 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, our assumption implies that � |Q|−1 � Q V s � 1 s ≤ C diam(Q)−1 for each (n − 1)-dimensional cube Q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Putting these facts together, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each (n − 1)-dimensional cube Q0 ∈ Q, we have � Q0 V |f − fQ0|2 ≤ C � Q0 Wg2, SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 23 where the function g : Q0 → R is defined by g(x) = sup Q∈Q,x∈Q⊂Q0 |Q|−1 � Q |f − fQ| for x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Fix an (n − 1)-dimensional cube Q0 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We define a function h : Q0 → R by h(x) = sup Q∈Q,x∈Q⊂Q0 |Q|−1 � Q |f − fQ0| for x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Note that V ≤ W and |f −fQ0| ≤ h at each point in Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Hence, it suffices to prove that � Q0 Wh2 ≤ C � Q0 Wg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' To prove this inequality, let α0 = |Q0|−1 � Q0 |f − fQ0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each α > α0, we denote by Qα the set of all (n − 1)-dimensional cubes Q ∈ Q with the following properties: Q ⊂ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' |Q|−1 � Q |f − fQ0| > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If ˜Q is an (n − 1)-dimensional cube in Q with Q ⊊ ˜Q and ˜Q ⊂ Q0, then | ˜Q|−1 � ˜Q |f − fQ0| ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' It is easy to see that |Q|−1 � Q |f − fQ0| ≤ 2n−1α for all Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In particular, |fQ − fQ0| ≤ 2n−1α for all Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, � Q∈Qα Q = {x ∈ Q0 : h(x) > α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Finally, no point can be contained in the interior of more than one cube in Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Let K > 1 and δ ∈ (0, 1) be two real numbers that will be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each (n−1)-dimensional cube Q ∈ Qα satisfying |Q|−1 � Q |f −fQ| ≤ δα, 24 SIMON BRENDLE we have (Kα − |fQ − fQ0|) � ˜Q∈QKα, ˜Q⊂Q | ˜Q| ≤ � ˜Q∈QKα, ˜Q⊂Q � � ˜Q |f − fQ0| − � ˜Q |fQ − fQ0| � ≤ � ˜Q∈QKα, ˜Q⊂Q � ˜Q |f − fQ| ≤ � Q |f − fQ| ≤ δα |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Recall that |fQ − fQ0| ≤ 2n−1α for all Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Hence, if we choose K > 2n, then we obtain � ˜Q∈QKα, ˜Q⊂Q | ˜Q| ≤ 21−n δ |Q| for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 � Q |f − fQ| ≤ δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We next apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='3 with ε = 1 2 K−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Hence, we can choose δ ∈ (0, 1) sufficiently small (depending on K) so that � ˜Q∈QKα, ˜Q⊂Q � ˜Q W ≤ 1 2 K−2 � Q W for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 � Q |f − fQ| ≤ δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each (n−1)-dimensional cube Q ∈ Qα, the set Q∩{h > Kα} is contained in the union � ˜Q∈QKα, ˜Q⊂Q ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This implies � Q∩{h>Kα} W ≤ 1 2 K−2 � Q W for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 � Q |f − fQ| ≤ δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' On the other hand, if Q is an (n − 1)-dimensional cube in Qα satisfying |Q|−1 � Q |f − fQ| > δα, then g > δα at each point in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Therefore, � Q∩{h>Kα} W ≤ � Q∩{g>δα} W for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 � Q |f − fQ| > δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Putting these facts together, we conclude that � Q∩{h>Kα} W ≤ 1 2 K−2 � Q W + � Q∩{g>δα} W SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 25 for each (n−1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Summation over all cubes Q ∈ Qα gives � {h>Kα} W ≤ 1 2 K−2 � {h>α} W + � {g>δα} W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This inequality holds for each α > α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, since g ≥ α0 at each point in Q0, the inequality is trivially true for α ≤ α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Finally, we multiply the inequality by α 2 and integrate over α ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This gives K−2 � Q0 Wh2 ≤ 1 2 K−2 � Q0 Wh2 + δ−2 � Q0 Wg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each (n − 1)-dimensional cube Q0 ∈ Q, we have � Q0 Wg2 ≤ C � Q0×[0,diam(Q)] |∇F|2, where the function g : Q0 → R is defined by g(x) = sup Q∈Q,x∈Q⊂Q0 |Q|−1 � Q |f − fQ| for x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Fix an (n−1)-dimensional cube Q0 ∈ Q, and let α0 = |Q0|−1 � Q0 |f− fQ0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each α > α0, we denote by Qα the set of all (n − 1)-dimensional cubes Q ∈ Q with the following properties: Q ⊂ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' |Q|−1 � Q |f − fQ| > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' If ˜Q is an (n − 1)-dimensional cube in Q with Q ⊊ ˜Q and ˜Q ⊂ Q0, then | ˜Q|−1 � ˜Q |f − f ˜Q| ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' It is easy to see that |Q|−1 � Q |f − fQ| ≤ 2nα for all Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, � Q∈Qα Q = {x ∈ Q0 : g(x) > α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Finally, no point can be contained in the interior of more than one cube in Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' 26 SIMON BRENDLE Let K > 1 be a real number that will be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each (n − 1)- dimensional cube Q ∈ Qα, we have Kα � ˜Q∈QKα, ˜Q⊂Q | ˜Q| ≤ � ˜Q∈QKα, ˜Q⊂Q � ˜Q |f − f ˜Q| ≤ 2 � ˜Q∈QKα, ˜Q⊂Q � ˜Q |f − fQ| ≤ 2 � Q |f − fQ| ≤ 2n+1α |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Hence, if we choose K > 2n+2, then � ˜Q∈QKα, ˜Q⊂Q | ˜Q| ≤ 1 2 |Q| for each cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' For each (n − 1)-dimensional cube Q ∈ Qα, the set Q ∩ {g > Kα} is contained in the union � ˜Q∈QKα, ˜Q⊂Q ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This implies |Q ∩ {g > Kα}| ≤ � ˜Q∈QKα, ˜Q⊂Q | ˜Q| ≤ 1 2 |Q|, hence |Q ∩ {g ≤ Kα}| ≥ 1 2 |Q| for each (n−1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' We define a nonnegative function ϕ : Rn−1 × {0} → R by ϕ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , xn−1, 0) = � � diam(Q0) 0 |∇F(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' , xn−1, xn)|2 dxn � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Moreover, we define a nonnegative function ψ : Q0 → R by ψ(x) = sup Q∈Q,x∈Q⊂Q0 |Q|−1 � Q ϕ for each point x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' In other words, ψ is the maximal function associated with ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using the Sobolev trace theorem, we obtain α ≤ |Q|−1 � Q |f − fQ| ≤ 2 |Q|−1 inf a∈R � Q |f − a| ≤ C |Q|−1 inf a∈R � � Q×[0,diam(Q)] |∇(F − a)| + diam(Q)−1 � Q×[0,diam(Q)] |F − a| � SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES 27 for each (n − 1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Using the Poincar´e inequality, we conclude that α ≤ C |Q|−1 � Q×[0,diam(Q)] |∇F| ≤ C diam(Q) 1 2 |Q|−1 � Q ϕ ≤ C diam(Q) 1 2 inf Q ψ for each (n − 1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' This implies α2 diam(Q)−1 |Q| ≤ C � Q∩{g≤Kα} ψ2 for each (n − 1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Combining this estimate with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content='4, we obtain α2 � Q W ≤ C � Q∩{g≤Kα} ψ2 for each (n−1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'} +page_content=' Summation over all cubes Q ∈ Qα gives α2 � {g>α} W ≤ � {α