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A rigidity theorem for asymptotically flat static manifolds and its applications

Brian Harvie, Ye-Kai Wang

Abstract

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds $(M^{n}, g)$ with boundary and with dimension $ n < 8$ that was establishedby McCormick. First, we show that any asymptotically flat static $(M^{n},g)$ which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems. As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension $n < 8$ under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes.

A rigidity theorem for asymptotically flat static manifolds and its applications

Abstract

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds with boundary and with dimension that was establishedby McCormick. First, we show that any asymptotically flat static which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems. As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes.
Paper Structure (14 sections, 21 theorems, 132 equations)

This paper contains 14 sections, 21 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Outline
  4. Preliminaries
  5. Asymptotically Flat Static Systems
  6. Inverse Mean Curvature Flow
  7. Outer-Minimizing Sets
  8. Umbilical Foliations by Inverse Mean Curvature Flow
  9. A Conformal Approach
  10. Applications
  11. Static Metric Extensions
  12. Equipotential CMC Spheres
  13. Photon Surfaces
  14. Hawking Mass Comparison

Key Result

Theorem 1.1

Let $(M^{n},g, V)$, $3 \leq n \leq 7$, be an asymptotically flat static system with ADM mass $m$ and compact, non-empty boundary $\partial M$. Suppose that $\partial M = \Sigma \cup (\cup_{i=1}^{k} \Sigma_{i})$, where $\Sigma$ is an outer-minimizing hypersurface and $\Sigma_{i}$ are closed minimal s on the boundary component $\Sigma$. Furthermore, equality holds in inequality if and only if $\part

Theorems & Definitions (50)

  • Theorem 1.1: static_minkowski, Static Minkowski Inequality
  • Theorem 1.2: Rigidity of Asymptotically Flat Static Manifolds
  • Remark 1.3
  • Theorem 1.4: Minkowski Inequality for Equipotential Boundaries
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7
  • Remark 1.8
  • Theorem 1.9: Uniqueness of Static Extensions for Constant, Schwarzschild-Stable Bartnik Data
  • Remark 1.10
  • ...and 40 more