Spectral Asymptotics of the Laplacian on Surfaces of Constant Curvature

The purpose of this paper is to explore the asymptotics of the eigenvalue spectrum of the Laplacian on 2 dimensional spaces of constant curvature, giving strong experimental evidence for a conjecture of the second author \cite{strichartz2016}. We computed and analyzed the eigenvalue spectra of several different regions in Euclidean, Hyperbolic, and Spherical space under Dirichlet, Neumann, and mixed boundary conditions and in particular we found that the average of the difference between the eigenvalue counting function and a 3-term prediction has the expected nice behavior. All computational code and data is available on our companion website \cite{murray2015results}