Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces

Abstract: In this work, we obtain the first upper bound on the multiplicity of Laplacian eigenvalues for negatively curved surfaces which is sublinear in the genus g. Our proof relies on a trace argument for the heat kernel, and on the idea of leveraging an r-net in the surface to control this trace. This last idea was introduced in [Jiang-Tidor-Yao-Zhang-Zhao, 2021] for similar spectral purposes in the context of graphs of bounded degree. Our method is robust enough to also yield an upper bound on the `approximate multiplicity'' of eigenvalues, i.e., the number of eigenvalues in windows of size $1/\log^\beta(g)$, $\beta\>0$. This work provides new insights on a conjecture by Colin de Verdi{\e}re [Colin de Verdi{`e}re, 1986] and new ways to transfer spectral results from graphs to surfaces.