Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mass, Conformal Capacity, and the Volumetric Penrose Inequality

Liam Mazurowski, Xuan Yao

Abstract

Let $Ω$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus Ω$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that $g$ has non-negative scalar curvature and that $Σ= \partial M$ is a minimal 2-sphere in the $g$ metric. We prove a sharp inequality relating the ADM mass of $M$ with the conformal capacity of $Ω$. As a corollary, we deduce a sharp lower bound for the ADM mass of $M$ in terms of the Euclidean volume of $Ω$. We also prove a stability type result for this ``volumetric Penrose inequality.'' The proofs are based on a monotonicity formula holding along the level sets of a 3-harmonic function.

Mass, Conformal Capacity, and the Volumetric Penrose Inequality

Abstract

Let be a smooth, bounded subset of diffeomorphic to a ball. Consider equipped with an asymptotically flat metric , where at infinity. Assume that has non-negative scalar curvature and that is a minimal 2-sphere in the metric. We prove a sharp inequality relating the ADM mass of with the conformal capacity of . As a corollary, we deduce a sharp lower bound for the ADM mass of in terms of the Euclidean volume of . We also prove a stability type result for this ``volumetric Penrose inequality.'' The proofs are based on a monotonicity formula holding along the level sets of a 3-harmonic function.

Paper Structure

This paper contains 17 sections, 21 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Background and Discussion
  3. Structure of Paper and Outline of Proof
  4. Acknowledgements
  5. Preliminaries
  6. Conformal Capacity
  7. Conformally Flat, Asymptotically Flat Manifolds
  8. The $U$ Function
  9. The Model Case
  10. The Smooth Setting
  11. Monotonicity in the Presence of Critical Points
  12. No critical points
  13. Monotonicity with negligible critical points
  14. An approximate monotonicity formula
  15. Asymptotics at Infinity
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\Omega\subset \mathbb{R}^3$ be a smooth, bounded domain diffeomorphic to a ball. Consider an asymptotically flat manifold $(M,g)$ where $M = \mathbb{R}^3\setminus \Omega$ and $g = f^4 g_{\text{euc}}$ for a smooth function $f > 0$ with $f \to 1$ at infinity. Assume that $M$ has non-negative scal Equality holds if and only if $M$ is Schwarzschild.

Theorems & Definitions (41)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4: Schwartz schwartz2011volumetric
  • Theorem 5: Freire and Schwartz freire2014mass
  • Remark 6
  • Definition 7
  • Example 8
  • Definition 9
  • Proposition 10
  • ...and 31 more