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Free boundary minimal surfaces in products of balls

Jaigyoung Choe, Ailana Fraser, Richard Schoen

TL;DR

The paper develops an extremal eigenvalue framework to construct free boundary minimal surfaces in products of Euclidean balls by linking the geometry to mixed Steklov-Neumann eigenvalues. It proves that the unrestricted maximum over all metrics does not exist in the product setting, and then leverages symmetry reduction and a canonical moduli space to obtain existence results: (i) genus-0 surfaces with six boundary components yield free boundary minimal immersions into rectangular prisms, and (ii) maximizing over conformal structures and symmetry groups produces free boundary minimal immersions into a product of balls. Central to the method is the functional $F(g)=\sum_{i=1}^k a_i^2 L_i(g)\sigma_1^{(i)}(g)$ and its odd/even decompositions, which tie eigenfunctions to coordinate maps into ball products via minimal immersions. The results provide a variational realization of Schwarz-type free boundary minimal surfaces in prisms and their generalizations in products of balls, with a detailed moduli-space analysis and symmetry-driven constructions.

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.

Free boundary minimal surfaces in products of balls

TL;DR

The paper develops an extremal eigenvalue framework to construct free boundary minimal surfaces in products of Euclidean balls by linking the geometry to mixed Steklov-Neumann eigenvalues. It proves that the unrestricted maximum over all metrics does not exist in the product setting, and then leverages symmetry reduction and a canonical moduli space to obtain existence results: (i) genus-0 surfaces with six boundary components yield free boundary minimal immersions into rectangular prisms, and (ii) maximizing over conformal structures and symmetry groups produces free boundary minimal immersions into a product of balls. Central to the method is the functional and its odd/even decompositions, which tie eigenfunctions to coordinate maps into ball products via minimal immersions. The results provide a variational realization of Schwarz-type free boundary minimal surfaces in prisms and their generalizations in products of balls, with a detailed moduli-space analysis and symmetry-driven constructions.

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.
Paper Structure (7 sections, 24 theorems, 144 equations)

This paper contains 7 sections, 24 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: mixed Steklov-Neumann problem
  3. Eigenvalue characterization of free boundary minimal submanifolds in products of balls
  4. The canonical model and the moduli space
  5. Symmetries of eigenfunctions
  6. Odd eigenvalues and minimal surfaces in $\mathbb R^3$
  7. Maximization and minimal surfaces in products of balls

Key Result

Proposition 2.1

Let $(M,g)$ be a Riemannian surface of genus $\gamma$ with $b$ boundary components, then where $C$ is independent of the metric $g$.

Theorems & Definitions (46)

  • Proposition 2.1: K, FS1, Ka
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 36 more