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Geometry of Einstein-type manifolds with boundary

Maria Andrade

TL;DR

The work investigates Einstein-type manifolds with boundary, a generalization of static spaces, and derives sharp geometric inequalities linking boundary geometry, spectral data of the Jacobi operator, and mass-type quantities. It proves a bound for the first Jacobi eigenvalue in terms of the boundary area and the boundary scalar curvature, and a complementary area bound governed by $\lambda_1$ and curvature data, with rigidity when equality occurs. A Brown–York mass-based boundary estimate is established for type-Einstein manifolds, yielding hemisphere rigidity under convex boundary embeddings, and a corollary expresses boundary area in terms of intrinsic boundary curvature. Collectively, these results extend static and quasi-Einstein inequalities to the boundary setting and highlight deep links between spectral geometry, curvature, and quasi-local mass.

Abstract

In this article, we consider Einstein-type manifolds with boundary which generalizes important geometric equations, like static vacuum and static perfect fluid. We investigate some geometric inequalities for those manifolds. Then, we established boundary estimates in terms of the first eigenvalue of the Jacobi operator and another one related to the Brown-York mass.

Geometry of Einstein-type manifolds with boundary

TL;DR

The work investigates Einstein-type manifolds with boundary, a generalization of static spaces, and derives sharp geometric inequalities linking boundary geometry, spectral data of the Jacobi operator, and mass-type quantities. It proves a bound for the first Jacobi eigenvalue in terms of the boundary area and the boundary scalar curvature, and a complementary area bound governed by and curvature data, with rigidity when equality occurs. A Brown–York mass-based boundary estimate is established for type-Einstein manifolds, yielding hemisphere rigidity under convex boundary embeddings, and a corollary expresses boundary area in terms of intrinsic boundary curvature. Collectively, these results extend static and quasi-Einstein inequalities to the boundary setting and highlight deep links between spectral geometry, curvature, and quasi-local mass.

Abstract

In this article, we consider Einstein-type manifolds with boundary which generalizes important geometric equations, like static vacuum and static perfect fluid. We investigate some geometric inequalities for those manifolds. Then, we established boundary estimates in terms of the first eigenvalue of the Jacobi operator and another one related to the Brown-York mass.
Paper Structure (7 sections, 10 theorems, 52 equations)

This paper contains 7 sections, 10 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of main results
  4. Proof of Theorem \ref{['estlam']}
  5. Proof of Theorem \ref{['boundlambda1']}
  6. Proof of the Theorem \ref{['massc']}
  7. Proof of Corollary \ref{['cmassc']}

Key Result

Theorem 1.1

Let $(M^3,g,f)$ be a compact oriented static triple with connected boundary $\partial M$ and scalar curvature $6.$ Then $\partial M$ is a two-sphere whose area satisfies the inequality which equality holding if and only if $M^n$ is isometric to the round hemisphere.

Theorems & Definitions (16)

  • Theorem 1.1: Boucher-Gibbons-Horowitz boucher1984uniqueness, Shen shen1997note
  • Theorem 1.2: andrade2024some
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1: andrade2024some
  • Lemma 2.2: coutinho2019static
  • Remark 2.3
  • Proposition 2.4
  • ...and 6 more