An annulus and a half-helicoid maximize Laplace eigenvalues

TL;DR

This paper proves that among surfaces of revolution and screw surfaces with the same boundary, the annulus and half-helicoid maximize the Dirichlet eigenvalues of the Laplace-Beltrami operator, using a sequence of geometric flattening arguments.

Contribution

It establishes the extremal property of the annulus and half-helicoid for Laplace eigenvalues among their respective classes of surfaces.

Findings

Annulus maximizes Laplace eigenvalues among revolution surfaces.

Half-helicoid maximizes Laplace eigenvalues among screw surfaces.

Flattening outside cylinders increases eigenvalues.

Abstract

The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in $\mathbb{R}^3$ with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinder successively increases the eigenvalues. A similar argument shows that the Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a half-helicoid than on any other screw surface in $\mathbb{R}^2 \times \mathbb{S}^1$ with the same boundary.