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The Riemannian Penrose Inequality with Matter Density

Hubert Bray, Yiyue Zhang

Abstract

Riemannian Penrose Inequalities are precise geometric statements that imply that the total mass of a zero second fundamental form slice of a spacetime is at least the mass contributed by the black holes, assuming that the spacetime has nonnegative matter density everywhere. In this paper, we remove this last assumption, and prove stronger statements that the total mass is at least the mass contributed by the black holes, plus a contribution coming from the matter density along the slice. We use the first author's conformal flow to achieve this, combined with Stern's harmonic level set techniques in the first case, and spinors in the second case. We then compare these new results to results previously known from Huisken-Ilmanen's inverse mean curvature flow techniques.

The Riemannian Penrose Inequality with Matter Density

Abstract

Riemannian Penrose Inequalities are precise geometric statements that imply that the total mass of a zero second fundamental form slice of a spacetime is at least the mass contributed by the black holes, assuming that the spacetime has nonnegative matter density everywhere. In this paper, we remove this last assumption, and prove stronger statements that the total mass is at least the mass contributed by the black holes, plus a contribution coming from the matter density along the slice. We use the first author's conformal flow to achieve this, combined with Stern's harmonic level set techniques in the first case, and spinors in the second case. We then compare these new results to results previously known from Huisken-Ilmanen's inverse mean curvature flow techniques.
Paper Structure (5 sections, 8 theorems, 58 equations, 2 figures)

This paper contains 5 sections, 8 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Conformal flow
  3. The Harmonic Level Set Formula
  4. Mass formula using spinors
  5. Formulae under conformal change of metrics

Key Result

Theorem 1.3

Let $(M^3,g)$ be a 3-dimensional smooth complete harmonically flat manifold. Define $m$ as the ADM mass of a chosen end and let $R$ be the scalar curvature of $(M^3,g)$. Suppose the outermost minimal surface of $(M^3,g)$ is nonempty and connected. Let $p_t$ and $q_t$ be two harmonic functions on $(M where $u_t$ and $\Sigma(t)$ are defined in Definition conflow, $\nu_t$ is the unit normal vector on

Figures (2)

  • Figure 1: An asymptotically flat manifold with a disconnected outermost minimal hypersurface $\Sigma_0$
  • Figure 2:

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • ...and 11 more