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Construction of GCM hypersurfaces in perturbations of Kerr

Dawei Shen

TL;DR

This work advances the nonlinear stability program for Kerr by removing symmetry restrictions in the general covariant modulated (GCM) framework and constructing GCM hypersurfaces in perturbations of Kerr. It develops a rigorous procedure to concatenate a 1-parameter family of GCM spheres into a spacelike GCM hypersurface via an ODE system for mode data, then enforces additional GCM conditions through a Banach fixed-point argument. The paper provides detailed geometric setup, relies on GCM spheres from KS Kerr1, and derives precise estimates for transition functions and extrinsic quantities to ensure controlled deformations. The resulting hypersurface Σ0 and its ODE governing Ψ,Λ,underline Λ constitute a key step toward a full nonlinear Kerr stability proof by enabling decay estimates without axial symmetry assumptions.

Abstract

This is a follow-up of \cite{KS:Kerr1} on the general covariant modulated (GCM) procedure in perturbations of Kerr. In this paper, we construct GCM hypersurfaces, which play a central role in extending GCM admissible spacetimes in \cite{KS:main} where decay estimates are derived in the context of nonlinear stability of Kerr family for $|a|\ll m$. As in \cite{KS}, the central idea of the construction of GCM hypersurfaces is to concatenate a $1$--parameter family of GCM spheres of \cite{KS:Kerr1} by solving an ODE system. The goal of this paper is to get rid of the symmetry restrictions in the GCM procedure introduced in \cite{KS} and thus remove an essential obstruction in extending the results to a full stability proof of the Kerr family.

Construction of GCM hypersurfaces in perturbations of Kerr

TL;DR

This work advances the nonlinear stability program for Kerr by removing symmetry restrictions in the general covariant modulated (GCM) framework and constructing GCM hypersurfaces in perturbations of Kerr. It develops a rigorous procedure to concatenate a 1-parameter family of GCM spheres into a spacelike GCM hypersurface via an ODE system for mode data, then enforces additional GCM conditions through a Banach fixed-point argument. The paper provides detailed geometric setup, relies on GCM spheres from KS Kerr1, and derives precise estimates for transition functions and extrinsic quantities to ensure controlled deformations. The resulting hypersurface Σ0 and its ODE governing Ψ,Λ,underline Λ constitute a key step toward a full nonlinear Kerr stability proof by enabling decay estimates without axial symmetry assumptions.

Abstract

This is a follow-up of \cite{KS:Kerr1} on the general covariant modulated (GCM) procedure in perturbations of Kerr. In this paper, we construct GCM hypersurfaces, which play a central role in extending GCM admissible spacetimes in \cite{KS:main} where decay estimates are derived in the context of nonlinear stability of Kerr family for . As in \cite{KS}, the central idea of the construction of GCM hypersurfaces is to concatenate a --parameter family of GCM spheres of \cite{KS:Kerr1} by solving an ODE system. The goal of this paper is to get rid of the symmetry restrictions in the GCM procedure introduced in \cite{KS} and thus remove an essential obstruction in extending the results to a full stability proof of the Kerr family.
Paper Structure (43 sections, 44 theorems, 512 equations, 2 figures)

This paper contains 43 sections, 44 theorems, 512 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stability of Kerr conjecture
  3. Stability of Schwarzschild in the polarized case
  4. GCM admissible spacetimes in
  5. The role played by GCM admissible spacetimes
  6. Review of the main results of
  7. Background space
  8. Null frame transformation
  9. Basis of modes
  10. Definition of GCM spheres
  11. Deformations of spheres and frame transformations
  12. GCM spheres with modes in
  13. First version of the main theorem
  14. Sketch of the proof of the main theorem
  15. Structure of the paper
  16. ...and 28 more sections

Key Result

Theorem 1.4

Let ${\mathcal{R}}$ be fixed spacetime region, endowed with an outgoing geodesic foliation $S(u, s)$, verifying specific asymptotic assumptionsCompatible with small perturbations of Kerr. expressed in terms of two parameters $0<\overset{\circ}{\delta}\leq \overset{\circ}{\epsilon}$. In particular we are small with respect to the parameter $\overset{\circ}{\delta}$. Let $\overset{\circ}{ S}=S(\over

Figures (2)

  • Figure 1: The GCM admissible space-time $\mathcal{M}$ of KS
  • Figure 2: The GCM hypersurface $\Sigma_0$ as a deformation of $\Sigma_\#=\{u=\Psi(s)\}$

Theorems & Definitions (113)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Existence of GCM spheres in KS:Kerr1
  • Remark 1.5
  • Theorem 1.6: Existence of GCM hypersurfaces, first version
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 103 more