Maximal spectral surfaces of revolution converge to a catenoid

TL;DR

This paper investigates the maximization of Laplace-Beltrami eigenvalues on revolution surfaces with fixed boundaries, showing convergence to a catenoid under certain conditions, revealing geometric and spectral relationships.

Contribution

It establishes the existence of maximizers for each eigenvalue and demonstrates convergence to a catenoid when it is the unique area-minimizing surface.

Findings

Maximizing surfaces have a meridian that is a rectifiable curve.

Existence of maximizers for each eigenvalue is proven.

Under certain conditions, these surfaces converge to a catenoid.

Abstract

We consider a maximization problem for eigenvalues of the Laplace-Beltrami operator on surfaces of revolution in $\mathbb{R}^3$ with two prescribed boundary components. For every $j$, we show that there is a surface $\Sigma_j$ which maximizes the $j$-th Dirichlet eigenvalue. The maximizing surface has a meridian which is a rectifiable curve. If there is a catenoid which is the unique area minimizing surface with the prescribed boundary, then the eigenvalue maximizing surfaces of revolution converge to this catenoid.