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Modified Hawking mass and rigidity of three-manifolds with boundary

Jihyeon Lee, Sanghun Lee

TL;DR

The paper proves a local rigidity result for 3-manifolds with boundary under a negative scalar curvature regime by examining a free boundary minimal two-disk that locally maximizes the modified Hawking mass. It develops a variational framework and sharp area estimates linked to the first Jacobi eigenvalue, then constructs a constant-mean-curvature foliation near the disk. Under the assumptions $\inf_{M} R^{M} = -6$ and $\inf_{\partial M} H^{\partial M} = 0$, the authors show the disk has constant Gaussian curvature $K^{\Sigma} = 1/a^{2}$ and vanishing boundary geodesic curvature, and the ambient space is locally isometric to the half anti-de Sitter–Schwarzschild manifold $(g_{hadss})_{a}$. This extends rigidity phenomena to manifolds with boundary in the AdS-like setting and connects quasi-local mass maximization to local geometric realizations in model spaces.

Abstract

In this paper, we prove a rigidity result for three-dimensional Riemannian manifolds with boundary, under the assumption that a free boundary minimal two-disk, which locally maximizes a modified Hawking mass, is embedded in a $3$-dimensional Riemannian manifold with negative scalar curvature and mean convex boundary. First, we establish area estimates for free boundary strictly stable two-disks. Finally, we show that the $3$-dimensional Riemannian manifold with boundary is locally isometric to the half anti-de Sitter-Schwarzschild manifold.

Modified Hawking mass and rigidity of three-manifolds with boundary

TL;DR

The paper proves a local rigidity result for 3-manifolds with boundary under a negative scalar curvature regime by examining a free boundary minimal two-disk that locally maximizes the modified Hawking mass. It develops a variational framework and sharp area estimates linked to the first Jacobi eigenvalue, then constructs a constant-mean-curvature foliation near the disk. Under the assumptions and , the authors show the disk has constant Gaussian curvature and vanishing boundary geodesic curvature, and the ambient space is locally isometric to the half anti-de Sitter–Schwarzschild manifold . This extends rigidity phenomena to manifolds with boundary in the AdS-like setting and connects quasi-local mass maximization to local geometric realizations in model spaces.

Abstract

In this paper, we prove a rigidity result for three-dimensional Riemannian manifolds with boundary, under the assumption that a free boundary minimal two-disk, which locally maximizes a modified Hawking mass, is embedded in a -dimensional Riemannian manifold with negative scalar curvature and mean convex boundary. First, we establish area estimates for free boundary strictly stable two-disks. Finally, we show that the -dimensional Riemannian manifold with boundary is locally isometric to the half anti-de Sitter-Schwarzschild manifold.
Paper Structure (4 sections, 6 theorems, 41 equations)

This paper contains 4 sections, 6 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Area estimates and CMC foliations
  4. Proof of Theorem \ref{['main thm']}

Key Result

Theorem 1.1

Let $M$ be a three-dimensional Riemannian manifold with boundary $\partial M$ satisfying $\inf_{M}R^{M} = -6$ and $\inf_{\partial M}H^{\partial M} = 0$. Suppose that $\Sigma$ is a properly embedded, two-sided, free boundary strictly stable minimal two-disk that locally maximizes the modified Hawking

Theorems & Definitions (6)

  • Theorem 1.1
  • Proposition 3.1: BBCBLSMN
  • Proposition 3.2: BBCBLSMN
  • Proposition 3.3
  • Proposition 3.4: BBCBLSMN
  • Lemma 4.1: Lemma 1 in BLS