An annulus and a half-helicoid maximize Laplace 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.