Geometric and spectral consequences of upper curvature bounds on planar graphs

We introduce a notion of curvature on general planar graphs to the study the geometrical and spectral consequences of upper curvature bounds: Assuming non-positive curvature already has strong implication on the local and global structure of the graph that is the graph is locally tessellating, infinite and every point has empty cut locus. Moreover, negative curvature yields empty minimal bigons, positive isoperimetric constants and positive exponential growth, where explicit upper bounds yield explicit estimates. For the spectrum this directly implies estimates on the bottom of the (essential) spectrum, where the case of uniformly decreasing curvature characterizes discrete spectrum and certain eigenvalue asymptotics. Finally, non positive curvature ensures that there are no eigenfunctions of compact support.