Math 762 — Spring 2002 Seminar in Geometry

This course is a continuation of Math 761.

The main topic of the course will be on conformal deformation of metrics on a compact manifold. I will show that any compact Riemannian manifold $(M^n,g)$ with boundary admits a metric of the form $e^{2f}g$ with constant scalar curvature and minimal boundary. This is known as the Yamabe problem on manifolds with boundary. Then I will study the problem of finding metrics with zero scalar curvature and prescribed mean curvature on the boundary. The above problems are equivalent to finding a solution to a nonlinear elliptic equation with critical Sobolev exponents. I will discuss different techniques in solving these kind of elliptic problems.

Other topics could include evolution equations such as Hamilton's Ricci flow and the mean curvature flow and/or the structure of manifolds with Ricci curvature bounded below.