Geometric and spectral consequences of curvature bounds on tessellations

TL;DR

This survey explores how curvature bounds influence the geometry and spectral properties of surface tessellations, highlighting both parallels and differences with Riemannian geometry.

Contribution

It provides a comprehensive overview of geometric and spectral consequences of curvature bounds in tessellations, including novel insights beyond classical Riemannian results.

Findings

Curvature bounds affect tessellation geometry and spectra.

Analogues and differences with Riemannian geometry are identified.

Results include both classical and new phenomena in discrete curvature.

Abstract

This is a chapter of a forthcoming Lecture Notes in Mathematics "Modern Approaches to Discrete Curvature" edited by L. Najman and P. Romon. It provides a survey on geometric and spectral consequences of curvature bounds. The geometric setting are tessellations of surfaces with finite and vanishing genus. We consider a curvature arising as an angular defect. Several of the results presented here have analogues in Riemannian geometry. In some cases one can go even beyond the Riemannian results and there also striking differences which shall be highlighted.