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A $σ$-homothetic uniqueness of the critical catenoid

Iury Domingos, Roney Santos, Feliciano Vitório

TL;DR

The paper addresses rigidity of free boundary minimal annuli in the unit ball that are $\sigma$-homothetic to the critical catenoid. It derives a differential system for the conformal factor between two free boundary minimal annuli, showing that $\Delta\varphi = (1 - e^{C - 2\varphi})K$ on $\Sigma$ with $\partial_\nu\varphi = e^{\varphi}$ on $\partial\Sigma$, and relates Gaussian curvatures via $e^{2\varphi}\bar K = K - \Delta\varphi$, yielding $4\varphi = C + \log(\bar K^{-1}K)$. The main result shows that if an annulus conformal to the critical catenoid has $\varphi$ constant on at least one boundary component, it is congruent to the critical catenoid; more generally, any free boundary annulus $\sigma$-homothetic to the critical catenoid is isometric to it. This advances rigidity results for free boundary minimal surfaces in $B^3$ and ties into the $\sigma$-homothety framework examined by Fraser–Schoen.

Abstract

We prove a uniqueness result for free boundary minimal annuli in the unit Euclidean three-ball that are $σ$-homothetic to the critical catenoid.

A $σ$-homothetic uniqueness of the critical catenoid

TL;DR

The paper addresses rigidity of free boundary minimal annuli in the unit ball that are -homothetic to the critical catenoid. It derives a differential system for the conformal factor between two free boundary minimal annuli, showing that on with on , and relates Gaussian curvatures via , yielding . The main result shows that if an annulus conformal to the critical catenoid has constant on at least one boundary component, it is congruent to the critical catenoid; more generally, any free boundary annulus -homothetic to the critical catenoid is isometric to it. This advances rigidity results for free boundary minimal surfaces in and ties into the -homothety framework examined by Fraser–Schoen.

Abstract

We prove a uniqueness result for free boundary minimal annuli in the unit Euclidean three-ball that are -homothetic to the critical catenoid.
Paper Structure (3 sections, 9 theorems, 17 equations)

This paper contains 3 sections, 9 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conformal changes and free boundary condition

Key Result

Theorem 1.1

Let $\Sigma$ and $\bar{\Sigma}$ be two conformal immersed free boundary minimal annuli in $B^3$ with conformal factor $\varphi \in C^4({\Sigma}).$ Then, there exists $C \in \mathbb{R}$ such that where $\Delta$ and $K$ are respectively the Laplacian and the Gaussian curvature of $\Sigma.$

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Lemma 2.1
  • Remark
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 6 more