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Quasi-spherical metrics and the static Minkowski inequality

Brian Harvie, Ye-Kai Wang

TL;DR

The paper proves a sharp rigidity result for equality in a static Minkowski-type inequality on asymptotically flat manifolds: equality occurs only for slices of Schwarzschild space. It develops a robust framework combining inverse mean curvature flow (IMCF) and quasi-spherical metrics, establishing smoothness of the weak IMCF after a finite time and deriving rotational symmetry of the metric and potential at infinity. Through detailed analysis of the static equations in polar coordinates and careful harmonic-mode expansions, it shows that equality enforces full rotational symmetry and canonical Schwarzschild geometry outside a compact set, with a precise metric form $g= u^{2} dr^{2} + r^{2} g_{S^{n-1}}$ and $V$ solving the static system. The results yield broad applications, including photon-surface uniqueness, static metric extension uniqueness, and higher-dimensional black hole uniqueness, all grounded in a boundary-energy framework and monotonicity of a static Minkowski functional. Overall, the work provides a dimension-agnostic rigidity criterion for Schwarzschild space in the static vacuum setting and advances understanding of quasi-local mass and boundary geometry in general relativity.

Abstract

We prove that equality within the Minkowski inequality for asymptotically flat static manifolds is achieved only by slices of Schwarzschild space.

Quasi-spherical metrics and the static Minkowski inequality

TL;DR

The paper proves a sharp rigidity result for equality in a static Minkowski-type inequality on asymptotically flat manifolds: equality occurs only for slices of Schwarzschild space. It develops a robust framework combining inverse mean curvature flow (IMCF) and quasi-spherical metrics, establishing smoothness of the weak IMCF after a finite time and deriving rotational symmetry of the metric and potential at infinity. Through detailed analysis of the static equations in polar coordinates and careful harmonic-mode expansions, it shows that equality enforces full rotational symmetry and canonical Schwarzschild geometry outside a compact set, with a precise metric form and solving the static system. The results yield broad applications, including photon-surface uniqueness, static metric extension uniqueness, and higher-dimensional black hole uniqueness, all grounded in a boundary-energy framework and monotonicity of a static Minkowski functional. Overall, the work provides a dimension-agnostic rigidity criterion for Schwarzschild space in the static vacuum setting and advances understanding of quasi-local mass and boundary geometry in general relativity.

Abstract

We prove that equality within the Minkowski inequality for asymptotically flat static manifolds is achieved only by slices of Schwarzschild space.
Paper Structure (21 sections, 46 theorems, 298 equations, 2 figures)

This paper contains 21 sections, 46 theorems, 298 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Applications of the main theorems
  3. Methods and structure of the paper
  4. Regularity of inverse mean curvature flow in asymptotically flat manifolds
  5. Quasi-spherical metrics and umbilical IMCF
  6. Static equations in polar coordinates
  7. The asymptotics of the metric coefficient $u$
  8. 3-dimensional case
  9. Higher-dimensional case
  10. Rotational symmetry to the order $r^{1-n}$
  11. Proof of Theorem \ref{['main']} and Theorem \ref{['equality case']}
  12. Rotational symmetry outside a compact set
  13. Proof of rotational symmetry and Theorem \ref{['main']}
  14. Proof of Theorem \ref{['equality case']}
  15. Applications
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $(M^{n},g,V)$ be asymptotically flat and let $\Sigma^{n-1} \subset \partial M$ be an outer-minimizing boundary component of $M$. Assume that Then we have the Minkowski-type inequality on the surface $\Sigma^{n-1}$.

Figures (2)

  • Figure 1: Two spheres in $\mathcal{A}$ with radius $s_1 < s_2$. The sphere $\partial B_{s_0}(0)$ is the horizon.
  • Figure 2: Space of arcs

Theorems & Definitions (89)

  • Theorem 1.1: The Static Minkowski Inequality
  • Theorem 1.2: Uniqueness of Quasi-Spherical Static Metrics
  • Remark
  • Theorem 1.3: Equality in the Static Minkowski Inequality
  • Theorem 1.4: Geometry and Uniqueness of Higher-Dimensional Static Vacuum Black Holes
  • Theorem 1.5: Geometry and Uniqueness of Higher-Dimensional Equipotential Photon Surfaces
  • Remark
  • Remark
  • Theorem 1.6: Uniqueness of Schwarzschild-Stable Static Metric Extensions
  • Theorem 2.1: Regularity of Weak IMCF in Asymptotically Flat Manifolds
  • ...and 79 more