PDEs in Conformal Geometry

In these lectures I will discuss two kinds of problems from conformal geometry, with the goal of showing an important connection between them in four dimensions. The first problem is a fully nonlinear version of the Yamabe problem, known as the k-Yamabe problem. This problem is, in general, not variational (or at least there is not a natural variational interpretation), and the underlying equation is second order but possibly not elliptic. Moreover, in contrast to the Yamabe problem, there is very little known (except for some examples and counterexamples) when the underlying manifold is negatively curved. © 2009 Springer-Verlag Berlin Heidelberg.