Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proof of the Spacetime Penrose Inequality With Suboptimal Constant in the Asymptotically Flat and Asymptotically Hyperboloidal Regimes

Brian Allen, Edward Bryden, Demetre Kazaras, Marcus Khuri

Abstract

We establish mass lower bounds of Penrose-type in the setting of $3$-dimensional initial data sets for the Einstein equations satisfying the dominant energy condition, which are either asymptotically flat or asymptotically hyperboloidal. More precisely, the lower bound consists of a universal constant multiplied by the square root of the minimal area required to enclose the outermost apparent horizon. Here the outermost apparent horizon may contain both marginally outer trapped (MOTS) and marginally inner trapped (MITS) components. The proof is based on the harmonic level set approach to the positive mass theorem, combined with the Jang equation and techniques arising from the stability argument of Dong-Song \cite{Dong-Song}. As a corollary, we also obtain a version of the Penrose inequality for 3-dimensional asymptotically hyperbolic Riemannian manifolds.

Proof of the Spacetime Penrose Inequality With Suboptimal Constant in the Asymptotically Flat and Asymptotically Hyperboloidal Regimes

Abstract

We establish mass lower bounds of Penrose-type in the setting of -dimensional initial data sets for the Einstein equations satisfying the dominant energy condition, which are either asymptotically flat or asymptotically hyperboloidal. More precisely, the lower bound consists of a universal constant multiplied by the square root of the minimal area required to enclose the outermost apparent horizon. Here the outermost apparent horizon may contain both marginally outer trapped (MOTS) and marginally inner trapped (MITS) components. The proof is based on the harmonic level set approach to the positive mass theorem, combined with the Jang equation and techniques arising from the stability argument of Dong-Song \cite{Dong-Song}. As a corollary, we also obtain a version of the Penrose inequality for 3-dimensional asymptotically hyperbolic Riemannian manifolds.

Paper Structure

This paper contains 8 sections, 17 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. The Generalized Exterior Region
  3. Energy, Harmonic Functions, and the Jang Graph
  4. Harmonic functions on the Jang graph
  5. Mass lower bounds on the Jang graph
  6. Dong-Song's Universal Poincaré-Faber-Szegö Inequality
  7. Proof of the Main Theorems
  8. Companion to Section \ref{['s:DSinequality']}

Key Result

Theorem 1.1

Let $(M,g,k)$ be a complete asymptotically flat initial data set satisfying the dominant energy condition. There exists a universal constant $\mathcal{C}<10^{18}$ such that where $m$ is the ADM mass of an end and $\mathcal{A}$ is the minimal area required to enclose an outermost apparent horizon associated with the end.

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Proposition 3.1
  • Remark 3.2
  • proof
  • ...and 30 more