Maximization of Laplace−Beltrami eigenvalues on closed Riemannian surfaces

Kao, Chiu-Yen; Lai, Rongjie; Osting, Braxton

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 685-720.

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 $(M,g)\to \lambda {}_k(M,g)$. We propose a computational method for finding the conformal spectrum ${\Lambda }_k^c(M,\left[g_0\right])$, which is defined by the eigenvalue optimization problem of maximizing $\lambda {}_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $\left[g_0\right]$ of fixed volume vol$(M,g)=1$. 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 ${\Lambda }_k^t\left(\gamma \right)$. Our computations support a conjecture of [N. Nadirashvili, J. Differ. Geom. 61 (2002) 335–340.] that ${\Lambda }_k^t\left(0\right)=8\pi k$, attained by a sequence of surfaces degenerating to a union of $k$ identical round spheres. Furthermore, based on our computations, we conjecture that ${\Lambda }_k^t\left(1\right)=\frac{8{\pi }^2}{\sqrt{3}}+8\pi (k-1)$, 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 ${\lambda }_k=4{\pi }^2{\lceil \frac{k}{2}\rceil }^2{({\lceil \frac{k}{2}\rceil }^2-\frac{1}{4})}^{-\frac{1}{2}}$. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.

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