Regularity Theory Of Short Geodesics In Space Of K?hler Metrics

In this paper,we first show the important role of geodesic in Kahler geometry.We describe the main difficulty and progress in the past,then state our theorem.We show that,given>4,0<J<min{1/4,k-4/4},any point in space of non-degenerate smooth Kahler potentials has a small neighborhood with w.r.t Ck norm so that any two points in this neighborhood can be connected by a geodesic of at least Ck-J regularity.To prove this we have to reprove Donaldson's result regarding stable solvability of Dirichlet Problem of HCMA equation on product space of Disc and manifold.The rea-son of doing this is that the original proof of Donaldson contains no apriori estimate.Our proof depends on Riemann-Hilbert problem on disc and also for the convience of proof we constructed weighted norm.Then we reduce the problem of looking for geodesic to the problem of looking for a fixed point of an iteration.The fixed point of this iteration would correspond to a geodesic.The iteration will be performed on a disc or more precisely a finite strip.To obtain the fixed point we have to apply Nash-Moser Type inverse function theorem,to compensate the loss of derivative of solution to HCMA equation.Finally we would construct examples to show our result is optimal in some sense.In the past,the tool of studying homogenous complex Monge-Ampere equation,is limited to the approximation method by nondegenerate elliptic equation or the method of upper envelop.The study of degenerate elliptic equations has its own difficulty and does not have complete results and the envelop method always provides only weak results,so for a long time the study of the geodesic equation is limited to lower order regularity.In this paper,by using the holomorphic leaf theory of Donaldson,we provide a higher order regularity result.This is a joint work with Prof.Xiuxiong Chen and Prof.Mikhail Feldman.