El problema de Yamabe

Abstract

[en] Posed by Hidehiko Yamabe in 1960, the Yamabe problem asks whether it is possible to deform the metric of a given riemannian manifold so that its scalar curvature becomes constant. This problem can be reformulated in terms of a partial differential equation which makes it interesting from an analytical point of view. In this work we aim to study the Yamabe problem in a variational way in order to find a solution when the scalar curvature is non-positive. To do so, we study Sobolev spaces and the critical value of the Rellich-Kondrakov embedding theorem toghether with its close connection with the solution of the Yamabe equation.