Free boundary minimal surfaces in products of balls

TL;DR

This paper introduces an extremal eigenvalue approach to construct free boundary minimal surfaces in products of Euclidean balls, revealing existence results and symmetry-based constructions.

Contribution

It develops a new extremal eigenvalue method for free boundary minimal surfaces in product spaces and demonstrates existence under symmetry constraints.

Findings

Constructed free boundary minimal surfaces in rectangular prisms.

Showed non-existence of an absolute maximum in the general case.

Identified maximizing metrics for genus zero surfaces with symmetries.

Abstract

In this paper we develop an extremal eigenvalue approach to the problem of construction of free boundary minimal surfaces in the product of Euclidean balls of chosen radii. The extremal problem involves a linear combination of normalized mixed Steklov-Neumann eigenvalues. The problem is motivated by the Schwarz P-surface which is a free boundary minimal surface in a cube. We show that the problem does not have an absolute maximum in the product case. By imposing a finite group of symmetries on both the surface and on the eigenfunctions we construct at least one free boundary minimal surface in a rectangular prism with arbitrary side lengths. We also show that for a genus zero surface with six boundary components and suitable reflection symmetries there is a maximizing metric which can be realized by a free boundary minimal immersion into a product of Euclidean balls.