An overview of curvature bounds and spectral theory of planar tessellations
We give a survey about the spectral consequences of upper bounds on the curvature on planar tessellating graphs. We first discuss spectral bounds and then put a particular focus on uniformly decreasing curvature. This case is characterized by purely discrete spectrum and we further present eigenvalue asymptotics and exponential decay of eigenfunctions. We then discuss absence of compactly supported eigenfunctions and dependence of the spectrum of the Laplacian on the underlying space.