Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces

Abstract

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by ${\lambda }_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M,g)\mapsto {\lambda }_k(M,g)$. We propose a computational method for finding the conformal spectrum ${\Lambda }_k^c(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing ${\lambda }_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $textrmvol(M,g)=1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfaces with fixed genus, $\gamma$. This is known as the topological spectrum for genus $\gamma$ and denoted by ${\Lambda }_k^t(\gamma )$. Our computations support a conjecture of N. Nadirashvili (2002) that ${\Lambda }_k^t(0)=8\pi k$, attained by a sequence of surfaces degenerating to a union of $k$ identical round spheres. Furthermore, based on our computations, we conjecture that ${\Lambda }_k^t(1)=\frac{8{\pi }^2}{\sqrt{3}}+8\pi (k-1)$, attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and $k-1$ identical round spheres. The values are compared to several surfaces where the Laplace-Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that among flat tori of volume one, the $k$-th Laplace-Beltrami eigenvalue has a local maximum with value ${\lambda }_k=4{\pi }^2{\lceil \frac{k}{2}\rceil }^2{\left({\lceil \frac{k}{2}\rceil }^2-\frac{1}{4}\right)}^{-\frac{1}{2}}$. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.

43 pages, 18 figures

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