Eigenvalue problems and free boundary minimal surfaces in spherical caps

TL;DR

This paper studies eigenvalue-based functionals on compact surfaces with boundary, characterizing maximizers as free boundary minimal immersions in spherical caps, and establishing symmetry properties of certain minimal surfaces.

Contribution

It introduces a new family of eigenvalue functionals, characterizes their maximizers as free boundary minimal surfaces, and proves symmetry results for minimal annuli in spherical caps.

Findings

Maximizers are induced by free boundary minimal immersions.

The disk maximizer is a spherical cap.

Free boundary minimal annuli are rotationally symmetric.

Abstract

Given a compact surface with boundary, we introduce a family of functionals on the space of its Riemannian metrics, defined via eigenvalues of a Steklov-type problem. We prove that each such functional is uniformly bounded from above, and we characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions.