Variational Methods in Geometry Seminar

It has been conjectured that a simply-connected complete Kahler manifold of negatively pinched sectional curvature is biholomorphic to a bounded domain in complex Euclidean space. One evidence is that the manifold is Stein, which is, in particular, a minimal submanifold in Euclidean space. Another evidence is that the manifold possesses invariant metrics. In 1979, R. E. Greene and H. Wu proved that the manifold possesses the complete Bergman metric, which dominates the background Kahler metric. Greene-Wu conjectured that the Bergman metric and the Kobayashi-Royden metric are both uniformly equivalent to the background Kahler metric. We shall present a proof of these conjectures and discuss some of the consequences. This is based on the joint work with S. T. Yau.