On Conformal Geometry of Kahler Surfaces

Abstract

In this thesis, we study several problems related to conformal geometry of Kahler and Einstein metrics on compact 4-manifolds, by using the conformally invariant Weyl functional. We first study a coupled system of equations on oriented compact 4-manifolds which we call the Bach-Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein-Maxwell equations. Inspired by the work of C. LeBrun on Einstein-Maxwell equations on compact Kahler surfaces, we give a variational characterization of solutions to Bach-Merkulov equations as critical points of the Weyl functional. We also show that extremal Kahler metrics are solutions to these equations, although, contrary to the Einstein-Maxwell analogue, they are not necessarily minimizers of the Weyl functional. We illustrate this phenomenon by studying the Calabi action on Hirzebruch surfaces. Next we prove that the only compact 4-manifold M with an Einstein metric of positive sectional curvature which is also hermitian with respect to some complex structure on M, is CP_2, with its Fubini-Study metric. Finally we present an alternative proof of existence of conformally compact Einstein metrics on some complex ruled surfaces fibered over Riemann surfaces of genus at least 2. This result was first proved by C. Tonnesen-Friedman. We prove the existence by finding the critical points of the Weyl functional on space of all extremal Kahler metrics on these ruled surfaces.