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Rigidity results for initial data sets satisfying the dominant energy condition

Christian Baer, Simon Brendle, Tsz-Kiu Aaron Chow, Bernhard Hanke

TL;DR

The paper proves rigidity theorems for initial data sets on compact spin manifolds with boundary and for compact convex polytopes under the dominant energy condition, by solving a boundary Dirac problem with chirality conditions. A twisted spinor bundle and a modified Weitzenböck identity yield nontrivial parallel spinors under the energy and boundary hypotheses, forcing geometric rigidities that realize the data as a spacelike hypersurface in Minkowski space with second fundamental form $q$. For polytopes, the authors approximate by smooth domains, impose a Matching Angle Hypothesis to control boundary errors, and obtain the analogous rigidity via a limiting immersion. Collectively, these results extend spacetime positive mass/rigidity phenomena to compact boundary settings and provide a spinorial route to convexity and embedding theorems in general relativity-inspired geometry.

Abstract

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary.

Rigidity results for initial data sets satisfying the dominant energy condition

TL;DR

The paper proves rigidity theorems for initial data sets on compact spin manifolds with boundary and for compact convex polytopes under the dominant energy condition, by solving a boundary Dirac problem with chirality conditions. A twisted spinor bundle and a modified Weitzenböck identity yield nontrivial parallel spinors under the energy and boundary hypotheses, forcing geometric rigidities that realize the data as a spacelike hypersurface in Minkowski space with second fundamental form . For polytopes, the authors approximate by smooth domains, impose a Matching Angle Hypothesis to control boundary errors, and obtain the analogous rigidity via a limiting immersion. Collectively, these results extend spacetime positive mass/rigidity phenomena to compact boundary settings and provide a spinorial route to convexity and embedding theorems in general relativity-inspired geometry.

Abstract

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary.
Paper Structure (15 sections, 73 theorems, 217 equations, 2 figures)

This paper contains 15 sections, 73 theorems, 217 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A boundary value problem for the Dirac operator in odd dimensions
  3. Spin geometry for initial data sets
  4. Proof of Theorem \ref{['rigidity.of.smooth.domains']}
  5. Proof of Proposition \ref{['prop:prelim']} in odd dimensions
  6. Proof of Proposition \ref{['prop:prelim']} in even dimensions
  7. Existence of a local isometric immersion into $\mathbb{R}^{n,1}$
  8. Proof of Theorem \ref{['rigidity.of.smooth.domains']} in the simply connected case
  9. Proof of Theorem \ref{['rigidity.of.smooth.domains']} in the general case
  10. Approximating a compact convex polytope by smooth domains
  11. Proof of Theorem \ref{['rigidity.of.polytopes']}
  12. Proof of Proposition \ref{['prop:polytope']} in odd dimensions
  13. Proof of Proposition \ref{['prop:polytope']} in even dimensions
  14. A metric characterization of the ball
  15. An auxiliary estimate

Key Result

Theorem I

Let $(M,g,q)$ be an initial data set. Assume that $M$ is a compact connected spin manifold, has dimension $n \geq 2$ and nonempty boundary $\partial M = \Sigma$. Let $N \colon \Sigma \to S^{n-1}$ be a smooth map. We assume that the following conditions are satisfied: Then $(M,g)$ is isometric to a manifold appearing in Example ex:model such that $q$ is the second fundamental form of $M$ in $\math

Figures (2)

  • Figure 1: The case $df(N)=0$ and $dN\ne0$
  • Figure 2: The case $df(N)\ne0$ and $dN=0$

Theorems & Definitions (153)

  • Example 1.1
  • Example 1.2
  • Theorem I
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem II
  • Remark 2.1
  • Lemma 2.2
  • ...and 143 more