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The Penrose inequality in extrinsic geometry

Michael Eichmair, Thomas Koerber

Abstract

The Riemannian Penrose inequality is a fundamental result in mathematical relativity. It has been a long-standing conjecture of G. Huisken that an analogous result should hold in the context of extrinsic geometry. In this paper, we resolve this conjecture and show that the exterior mass $m$ of an asymptotically flat support surface $S\subset\mathbb{R}^3$ with nonnegative mean curvature and outermost free boundary minimal surface $D$ is bounded in terms of $$ m\geq \sqrt{\frac{|D|}π}. $$ If equality holds, then the unbounded component of $S\setminus \partial D$ is a half-catenoid. In particular, this extrinsic Penrose inequality leads to a new characterization of the catenoid among all complete embedded minimal surfaces with finite total curvature. To prove this result, we study minimal capillary surfaces supported on $S$ that minimize the free energy and discover a quantity associated with these surfaces that is nondecreasing as the contact angle increases.

The Penrose inequality in extrinsic geometry

Abstract

The Riemannian Penrose inequality is a fundamental result in mathematical relativity. It has been a long-standing conjecture of G. Huisken that an analogous result should hold in the context of extrinsic geometry. In this paper, we resolve this conjecture and show that the exterior mass of an asymptotically flat support surface with nonnegative mean curvature and outermost free boundary minimal surface is bounded in terms of If equality holds, then the unbounded component of is a half-catenoid. In particular, this extrinsic Penrose inequality leads to a new characterization of the catenoid among all complete embedded minimal surfaces with finite total curvature. To prove this result, we study minimal capillary surfaces supported on that minimize the free energy and discover a quantity associated with these surfaces that is nondecreasing as the contact angle increases.

Paper Structure

This paper contains 12 sections, 48 theorems, 183 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Sub- and supersolutions
  3. Minimizing the free energy
  4. Stable minimal capillary surfaces minimizing the free energy
  5. Monotonicity of the free energy mass
  6. Asymptotic behavior of the free energy mass
  7. Proofs of Theorem \ref{['extrinsic:rpi']} and Theorem \ref{['extrinsic:pmt']}
  8. Embedded minimal surfaces with finite total curvature
  9. Asymptotically flat support surfaces
  10. First and second variation of area and enclosed area
  11. The free energy and minimal capillary surfaces
  12. Sets of finite perimeter

Key Result

Theorem 1

Let $S\subset\mathbb{R}^3$ be an asymptotically flat support surface with nonnegative mean curvature. There holds $m\geq 0$ with equality if and only if $S$ is a flat plane.

Figures (4)

  • Figure 1: An illustration of the asymptotically flat support surface $S_m$. The outermost free boundary minimal surface $D_m$ is indicated by the solid gray line. The exterior surface $S_m'$ is indicated by the dashed black line. The exterior region $M'(S_m)$ is indicated by the hatched region.
  • Figure 2: An illustration of an admissible surface $\Sigma$ indicated by the solid gray line with two components supported on an asymptotically flat support surface $S$ indicated by the solid black line. The lateral surface $S(\Sigma)$ of $\Sigma$ is indicated by the dashed black line. The inside $\Omega(\Sigma)$ of $\Sigma$ is indicated by the hatched region.
  • Figure 3: An illustration of the calibration argument used in the proof of Lemma \ref{['replacement lemma']}. $S$ is indicated by the solid black line. $\Omega(\Sigma_{t_0})$ is indicated by the hatched region. $\partial F_k$ is indicated by the solid gray line and $\{x\in \bar{M}(S):v(x)=t\}$ is indicated by the dashed gray line. The divergence theorem is applied to the region indicated by the gray filling.
  • Figure 4: An illustration of the proof of Lemma \ref{['inner divergence']}. $\Psi^{-1}(\partial \Sigma_{t_0})$ has three components, which are illustrated by the dashed black lines. $\Psi^{-1}(\partial \Sigma_{t_k})$ has two components, which are illustrated by the solid black lines. $\Gamma_k$, which is illustrated by the solid gray line, touches one of these two components at $y_k$.

Theorems & Definitions (96)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Theorem 10
  • ...and 86 more