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 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^β(g), β>0. This work provides new insights on a conjecture by Colin de Verdière and new ways to transfer spectral results from graphs to surfaces.

在这项工作中，我们获得了负曲面拉普拉斯特征值重数的第一个上限，该曲面在属 g 中是次线性的。我们的证明依赖于热内核的跟踪参数，以及利用表面中的 r-net 来控制该跟踪的想法。最后一个想法于 2021 年引入，用于有界度图背景下的类似光谱目的。我们的方法足够稳健，还可以产生特征值“近似重数”的上限，即大小为 1/log ^β (g) 的窗口中特征值的数量，β>0。这项工作为 Colin de Verdière 的猜想提供了新的见解，以及将光谱结果从图转移到表面的新方法。