Le probl\'eme de Yamabe avec singularit\'es et la conjecture de Hebey-Vaugon

Abstract

In the first part of this thesis, we study the Yamabe problem with singularities, that we can announce as follow: Given a compact Riemannian manifold $(M,g)$, find a constant scalar curvature metric, conformal to $g$, when $g$ has not necessarily the usual regularity (it can be $C^1$). To solve this problem, we start the study of the Yamabe type equations. We show that all the known properties in the smooth case are still valid. Under some assumptions, we prove the existence and uniqueness of solutions. The second part is dedicated to the Hebey-Vaugon conjecture, stated in their paper about the equivariant Yamabe problem. We prove that this conjecture is true in some new cases, after we generalize T. Aubin's theorem.