On non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics

TL;DR

This paper proves the non-degeneracy of the linearised Einstein operator for a class of Riemannian Schwarzschild-anti de Sitter metrics, with specific exceptions related to the mass parameter and topology.

Contribution

It establishes non-degeneracy results for the linearised Einstein operator on Riemannian Schwarzschild-anti de Sitter metrics, highlighting conditions based on dimension, topology, and mass.

Findings

Non-degeneracy holds for most such metrics.

Exceptions occur for spherical metrics with critical mass.

Provides a detailed analysis of the mass parameter's role.

Abstract

We prove that the $TT$-gauge-fixed linearised Einstein operator is non-degenerate for Riemannian Kottler ("Schwarzschild-anti de Sitter") metrics with dimension- and topology-dependent ranges of mass parameter. We provide evidence that this remains true for all such metrics except the spherical ones with a critical mass.