Local rigidity for Yamabe-type problems in warped products

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.

24 pages, 2 figures. Thoroughly revised version

References in corpus (2)

• Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions
• Rigidity and bifurcation results for CMC hypersurfaces in warped product spaces