On some conformally invariant fully nonlinear equations, II. Liouville, Harnack and Yamabe

Abstract

Let (M, g) be an n-dimensional compact smooth Riemannian manifold (without boundary). For n=2, we know from the uniformization theorem of Poinca% that there exist metrics that are pointwise conformal to g and have constant Gauss curvature. For n~>3, the well-known Yamabe conjecture states that there exist metrics which are pointwise conformal to g and have constant scalar curvature. The Yamabe conjecture is proved through the work of Yamabe [65], Trudinger [58], Aubin [2] and Schoen [53]. The Yamabe and related problems have attracted much attention in the last 30 years or so, see, e.g., [57], [3] and the references therein. Important methods and techniques in overcoming loss of compactness have been developed in such studies, which also play important roles in the research of other areas of mathematics. For n~>3, let ~=u4/(n-2)g, where u is some positive function on M. The scalar curvature RO of ~ can be calculated as