Sectional curvature of polygonal complexes with planar substructures

TL;DR

This paper introduces a class of polygonal complexes with a notion of sectional curvature, generalizing Euclidean and hyperbolic buildings, and explores their geometric, spectral, and analytical properties.

Contribution

It defines sectional combinatorial curvature for polygonal complexes and establishes key geometric and spectral results, including hyperbolicity and eigenvalue estimates.

Findings

Proves Gromov hyperbolicity for complexes with negative curvature

Establishes spectral discreteness and eigenvalue asymptotics

Demonstrates unique continuation and Dirichlet problem solvability

Abstract

In this paper we introduce a class of polygonal complexes for which we can define 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.