Elliptic and parabolic problems in conformal geometry

Abstract

I will review recent results regarding two questions that arise in connection with the Yamabe problem. The first problem is concerned with the conformal deformation of Riemannian metrics by their scalar curvature. This leads to a parabolic evolution equation, which can be interpreted as the flow of steepest descent for the Yamabe functional. I will provide conditions which guarantee that the flow converges to a metric of constant scalar curvature as _t_→∞. The second problem is concerned with the set of constant scalar curvature metrics in a given conformal class. I will discuss under what conditions this set is compact. A recurring theme in the study of both problems is that blow-up can be ruled out by means of the positive mass theorem provided that the dimension is less than 6, whereas the Weyl tensor plays a crucial role in higher dimensions.