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Stability and rigidity of axisymmetric marginally outer trapped two-spheres

Gregory J. Galloway, Abraão Mendes

TL;DR

The article develops a rigorous axisymmetric analogue of rigidity results for MOTS in three-dimensional initial data sets by combining the MOTS stability operator with Killing-symmetry techniques. It proves that axisymmetric, stable MOTS invariant under a nontrivial Killing field satisfy a sharp area bound $| riangle|\,igackslashig| rac{4 ext{π}}{c+oldsymbol{ ext{ω}}}ig.$, where $oldsymbol{ ext{ω}}$ encodes angular momentum via $|X^{ ext{η}}|^2$, and identifies the rigidity case where a local product splitting occurs. A foliation by axisymmetric constant outward null expansion surfaces is established, with injectivity and surjectivity of the foliation map ensuring a local splitting structure near $ riangle$. The work further scrutinizes rotating Nariai spacetimes, illustrating how angular momentum affects area bounds and rigidity, and shows that stationary regions admit foliations by MOTS with vanishing outward null expansion. Overall, this advances understanding of how symmetry and energy conditions constrain the geometry of MOTS in GR initial data and its interplay with angular momentum.

Abstract

In [7], H. Bray, S. Brendle, and A. Neves studied rigidity properties of area-minimizing two-spheres in Riemannian three-manifolds with uniformly positive scalar curvature. In [13], these results were extended to marginally outer trapped surfaces (MOTS) in general initial data sets $(M^3,g,K)$ under a natural energy condition. In the present work, we refine the latter results to the setting of axisymmetric MOTS in initial data sets admitting a nontrivial Killing vector field. Conditions for the stability of such MOTS, as well as a new foliation lemma by axisymmetric surfaces of constant outward null expansion, are obtained. Finally, we discuss some aspects of the rotating Nariai spacetimes and their relation to these results.

Stability and rigidity of axisymmetric marginally outer trapped two-spheres

TL;DR

The article develops a rigorous axisymmetric analogue of rigidity results for MOTS in three-dimensional initial data sets by combining the MOTS stability operator with Killing-symmetry techniques. It proves that axisymmetric, stable MOTS invariant under a nontrivial Killing field satisfy a sharp area bound , where encodes angular momentum via , and identifies the rigidity case where a local product splitting occurs. A foliation by axisymmetric constant outward null expansion surfaces is established, with injectivity and surjectivity of the foliation map ensuring a local splitting structure near . The work further scrutinizes rotating Nariai spacetimes, illustrating how angular momentum affects area bounds and rigidity, and shows that stationary regions admit foliations by MOTS with vanishing outward null expansion. Overall, this advances understanding of how symmetry and energy conditions constrain the geometry of MOTS in GR initial data and its interplay with angular momentum.

Abstract

In [7], H. Bray, S. Brendle, and A. Neves studied rigidity properties of area-minimizing two-spheres in Riemannian three-manifolds with uniformly positive scalar curvature. In [13], these results were extended to marginally outer trapped surfaces (MOTS) in general initial data sets under a natural energy condition. In the present work, we refine the latter results to the setting of axisymmetric MOTS in initial data sets admitting a nontrivial Killing vector field. Conditions for the stability of such MOTS, as well as a new foliation lemma by axisymmetric surfaces of constant outward null expansion, are obtained. Finally, we discuss some aspects of the rotating Nariai spacetimes and their relation to these results.

Paper Structure

This paper contains 10 sections, 10 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stability and infinitesimal rigidity of axisymmetric MOTS
  4. Case 1: $\Sigma^2$ is a two-torus.
  5. Case 2: $\Sigma^2$ is a two-sphere.
  6. Foliation lemma
  7. Injectivity.
  8. Surjectivity.
  9. Main result: Local splitting
  10. Nariai spacetime with rotation

Key Result

Lemma 1

Let $(\Sigma^n,g_\Sigma)$ be a closed Riemannian manifold, and let $\eta$ be a Killing vector field on $\Sigma^n$. If $X \in \mathfrak{X}(\Sigma)$ and $Q \in C^\infty(\Sigma)$ are invariant under $\eta$, then the principal eigenfunction of the operator is also invariant under $\eta$.

Theorems & Definitions (25)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Remark 4
  • Proposition 5: Infinitesimal rigidity
  • proof
  • Remark 6
  • ...and 15 more