Local rigidity for Yamabe-type problems in warped products

TL;DR

This paper investigates conditions under which metrics with constant scalar and mean curvature on manifolds with boundary are locally rigid, introducing a new spectral problem and applying results to specific spacetime slices.

Contribution

It introduces geometrical conditions ensuring local rigidity for Yamabe-type problems, based on a novel spectral problem called the 'mixed eigenvalue problem,' and applies these to spacetime models.

Findings

Established geometrical hypotheses for local rigidity.

Defined and studied a new spectral problem beyond classical ones.

Applied results to de Sitter and Anti-de Sitter Schwarzschild spacetimes.

Abstract

Our aim in this paper is to study local rigidity for metrics defined on a compact manifold $M$ with boundary satisfying constant scalar curvature on $M$ and constant mean curvature on $\partial M$. We present some geometrical hypotheses ensuring local rigidity for both, the general Riemannian and the warped metric case. These conditions arise from the study of a spectral problem which is not included within the classical problems (Neumann, Steklov,...) that we call "mixed eigenvalue problem". Finally, we apply our previous results for the spatial slice of the de Sitter and Anti-de Sitter Schwarzschild spacetimes.