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arXiv · 1912.12195

Effective results on uniformization and intrinsic GCM spheres in perturbations of Kerr

Sergiu Klainerman·Jeremie Szeftel
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Abstract

This is a follow-up of our paper \cite{KS-Kerr1} on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the main results of that paper using a canonical definition of $\ell=1$ modes on a $2$-sphere embedded in a $1+3$ vacuum manifold. This is based on a new, effective, version of the classical uniformization theorem which allows us to define such modes and prove their stability for spheres with comparable metrics. The reformulation allows us to prove a second, intrinsic, existence theorem for GCM spheres, expressed purely in terms of geometric quantities defined on it. A natural definition of angular momentum for such GCM spheres is also introduced, which we expect to play a key role in determining the final angular momentum for general perturbations of Kerr.

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Sergiu Klainerman, Jeremie Szeftel. 2019-12-27. Effective results on uniformization and intrinsic GCM spheres in perturbations of Kerr. https://arxiv.org/abs/1912.12195

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