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Riemannian Penrose inequality via Nonlinear Potential Theory

Virginia Agostiniani, Carlo Mantegazza, Lorenzo Mazzieri, Francesca Oronzio

TL;DR

The paper proves the Riemannian Penrose inequality for time-symmetric, asymptotically flat 3-manifolds with a single horizon by replacing the inverse mean curvature flow with a level-set flow of the $p$-capacitary potential. It develops a monotone quantity $F_p(t)$ along regular level sets, derived from a nonlinear potential-theoretic construction and a Geroch-type computation, and extends the monotonicity to settings with critical points via approximation and Sard-type arguments. The main results include a complete proof for the single-black-hole case and a robust framework that handles singular level sets through a regularization scheme, ultimately connecting the horizon geometry to the ADM mass. This approach highlights nonlinear potential theory as a viable alternative to curvature flows for establishing geometric inequalities in general relativity, with potential for broader applicability.

Abstract

We provide a new proof of the Riemannian Penrose inequality for time-symmetric asymptotically flat initial data with a single black-hole horizon. The proof proceeds through a newly established monotonicity formula holding along the level sets of the $p$-capacitary potential of the horizon boundary, in any asymptotically flat $3$-manifold with nonnegative scalar curvature.

Riemannian Penrose inequality via Nonlinear Potential Theory

TL;DR

The paper proves the Riemannian Penrose inequality for time-symmetric, asymptotically flat 3-manifolds with a single horizon by replacing the inverse mean curvature flow with a level-set flow of the -capacitary potential. It develops a monotone quantity along regular level sets, derived from a nonlinear potential-theoretic construction and a Geroch-type computation, and extends the monotonicity to settings with critical points via approximation and Sard-type arguments. The main results include a complete proof for the single-black-hole case and a robust framework that handles singular level sets through a regularization scheme, ultimately connecting the horizon geometry to the ADM mass. This approach highlights nonlinear potential theory as a viable alternative to curvature flows for establishing geometric inequalities in general relativity, with potential for broader applicability.

Abstract

We provide a new proof of the Riemannian Penrose inequality for time-symmetric asymptotically flat initial data with a single black-hole horizon. The proof proceeds through a newly established monotonicity formula holding along the level sets of the -capacitary potential of the horizon boundary, in any asymptotically flat -manifold with nonnegative scalar curvature.
Paper Structure (7 sections, 6 theorems, 112 equations)

This paper contains 7 sections, 6 theorems, 112 equations.

Table of Contents

  1. A monotonicity formula for $p$--capacitary potentials.
  2. Setting and preliminaries.
  3. Monotonicity without critical points.
  4. Monotonicity with negligible critical values.
  5. An approximate monotonicity formula.
  6. Proof of Theorem \ref{['thm:monotonicity']}.
  7. Proof of the Riemannian Penrose inequality for a single black hole

Key Result

Theorem 1.1

Let $(M,g)$ be a $3$--dimensional, complete, noncompact Riemannian manifold with nonnegative scalar curvature and smooth, compact and connected boundary $\partial M$. Moreover, assume that $H_2(M, \partial M;\mathbb{Z})=\{0\}$ and suppose that, for every $1<p<3$, problem f1 admits a unique solution where $F_p$ is the function defined in formula defFp.

Theorems & Definitions (13)

  • Theorem 1.1: Monotonicity along the regular values
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Definition 2.1
  • Theorem 2.2: Riemannian Penrose inequality for a single black hole
  • Remark 2.3
  • proof : Proof of Theorem \ref{['RPI']}
  • Lemma 2.4
  • ...and 3 more