| Let H be the space of Kahler metrics in a fixed cohomology class. This space may be endowed with a Weil-Petersson-type metric, referred to as the Mabuchi metric, which allows one to study the geometry of H. It is now well-known that the geometry of the space of Kahler potentials, in particular, the geodesics in H, may be used for studying 'canonical metrics' on the base manifold. In order to be interpreted as the potential of a Kahler metric, however, one needs to prove certain regularity for such solutions.;In the first part, I shall discuss deriving of weighted estimates for the space and time derivatives of solutions in the case of ALE Kahler potentials, and further, prove results regarding the Mabuchi energy and the uniqueness of metrics of constant scalar curvature. In the latter part of the talk I will discuss certain weighted estimates for the solutions to the geodesic equation when the end points have conical singularities. The results may also be seen as X.-X. Chen's fundamental work on the geodesic convexity of H in the case of smooth compact manifolds. |
