Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces

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 $1\mathrm{/}{\log }^{\beta }(g)$, $\beta >0$. This work provides new insights on a conjecture by Colin de Verdi{è}re [Colin de Verdi{è}re, 1986] and new ways to transfer spectral results from graphs to surfaces.