Duong Hong Phong

Abstract:   The Kahler-Ricci flow is a parabolic flow of metrics, whose fixed points are Kahler-Einstein metrics. According to a well- known conjecture of Yau, the existence of such metrics, and hence the convergence of the flow, should be equivalent to the stability of the underlying manifold in the sense of geometric invariant theory. We discuss recent progresses in this direction, including conditions for the convergence of the flow in terms of orbits of almost-complex structure under the diffeomorphism group, and lower bounds for the $\bar\partial$-operator on vector fields.