The prescribed Mean Curvature Problem on four dimensional manifolds with boundary

Abstract

In this thesis we study the prescribed Mean Curvature Problem (PMCP) on four-dimensional Riemannian Manifolds with boundary. Given a smooth function K on the boundary, the PMCP asks for conditions on K, such that the given metric is conformally equivalent to a metric with mean curvature given by K and vanishing scalar curvature in the interior.This problem is equivalent to finding critical points of a given functional, which unfor tunately does not satisfy the Palais-Smale condition. We use the method of "Critical points at infinity", developed by Abbas Bahri, to study non-converging flow lines of a suitable pseudo gradient vector field. We understand "limit sets" of those flow lines and understand the difference of topology in the variational space, induced by these flow lines.Comparing this difference of topology to the topology of the variational space yields existence results for critical points of the given functional. And therefore conditions on K such that K can be realized as the mean curvature of a conformal metric.