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Free boundary minimal surfaces and the reflection principle

Jaigyoung Choe

TL;DR

This work generalizes Schwarz's reflection principle to a sphere, proving that a minimal surface meeting the unit sphere orthogonally can be analytically reflected across the sphere. By solving a Cauchy problem for the Laplacian, the authors obtain isothermal coordinates with $F(X,0)=1$ on the free boundary, transforming the Steklov condition into Schwarz-type data. Repeated reflections across the boundary components produce a complete minimal surface with two ends, which is shown to be the catenoid via curvature and Gauss-map analysis, thereby proving that any embedded free boundary minimal annulus in a ball must be the critical catenoid. This result resolves a longstanding conjecture for embedded annuli and highlights the power of a sphere-based reflection principle in free boundary minimal surface theory.

Abstract

We show that a minimal surface meeting a sphere at a 90-degree angle can be reflected across the sphere. Using this reflection, we prove the uniqueness that every embedded free boundary minimal annulus in a ball is necessarily the critical catenoid.

Free boundary minimal surfaces and the reflection principle

TL;DR

This work generalizes Schwarz's reflection principle to a sphere, proving that a minimal surface meeting the unit sphere orthogonally can be analytically reflected across the sphere. By solving a Cauchy problem for the Laplacian, the authors obtain isothermal coordinates with on the free boundary, transforming the Steklov condition into Schwarz-type data. Repeated reflections across the boundary components produce a complete minimal surface with two ends, which is shown to be the catenoid via curvature and Gauss-map analysis, thereby proving that any embedded free boundary minimal annulus in a ball must be the critical catenoid. This result resolves a longstanding conjecture for embedded annuli and highlights the power of a sphere-based reflection principle in free boundary minimal surface theory.

Abstract

We show that a minimal surface meeting a sphere at a 90-degree angle can be reflected across the sphere. Using this reflection, we prove the uniqueness that every embedded free boundary minimal annulus in a ball is necessarily the critical catenoid.
Paper Structure (5 sections, 10 theorems, 97 equations)

This paper contains 5 sections, 10 theorems, 97 equations.

Table of Contents

  1. introduction
  2. Cauchy problem
  3. reflection across a sphere
  4. repeated reflections
  5. critical catenoid

Key Result

Lemma 2.1

Let's define $\Sigma,x,y,F(x,y),\Psi,D$, $\delta$, $\gamma$ and $\nu$ as above. Then there exists a superharmonic function $k$ on $\Sigma$ satisfying the Cauchy conditions

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • proof
  • ...and 10 more