Maximal spectral surfaces of revolution converge to a catenoid

Abstract

Abstract. We consider a maximization problem for eigenvalues of the LaplaceBeltrami operator on surfaces of revolution in R 3 with two prescribed boundary components. For every j, we show there is a surface Σ 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.