On Harnack inequalities and singularities of admissible metrics in the Yamabe problem

Abstract

In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds $(M, g_0)$ of dimension $n\ge 3$. For $n/2 <k<n$, we prove a sharp Harnack inequality for admissible metrics when $(M,g_0)$ is not conformally equivalent to the unit sphere $S^n$ and that the set of all such metrics is compact. When $(M,g_0)$ is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering a recent result of Gursky and Viaclovski on the solvability of the $k$-Yamabe problem for $k>n/2$.