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

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 ≥ 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 as a special case, a recent result of Gursky and Viaclovsky on the solvability of the k-Yamabe problem for k > n/2.