| In 1997, Tian proved the Moser-Trudinger type inequality on compact Kahler-Einstein manifolds which played an important role in the study of Kahler-Einstein geometry. We generalize this result to compact Kahler manifolds with multiplier Hermitian structures and define the generalized Futaki invariant to such manifolds.; We also study the boundary value problem of the complex Monge-Ampere equation on bounded domains in Cn. We use the heat flow method to show the long time existence of the parabolic Monge-Ampere equation and define a functional F associated to the equation to show the convergence of the flow. |
