Sectional curvature of polygonal complexes with planar substructures

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• 作者：Matthias Keller^a ; ^{m.keller@uni-jena.de} ; Norbert Peyerimhoff^b ; ^{norbert.peyerimhoff@durham.ac.uk} ; Felix Pogorzelski^c ; ^{felixp@technion.ac.il}
• 关键词：Buildings ; Cheeger constant ; Combinatorial curvature ; Compactly supported eigenfunctions ; Dirichlet problem at infinity ; Graph Laplacians ; Gromov hyperbolicity
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：307
• 期：Complete
• 页码：1070-1107
• 全文大小：687 K
• 卷排序：307

文摘

In this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature. These complexes can be viewed as generalizations of 2-dimensional Euclidean and hyperbolic buildings. We focus on the case of non-positive and negative combinatorial curvature. As geometric results we obtain a Hadamard–Cartan type theorem, thinness of bigons, Gromov hyperbolicity and estimates for the Cheeger constant. We employ the latter to get spectral estimates, show discreteness of the spectrum in the sense of a Donnelly–Li type theorem and present corresponding eigenvalue asymptotics. Moreover, we prove a unique continuation theorem for eigenfunctions and the solvability of the Dirichlet problem at infinity.