4/16/2010

Slawomir Dinew
Uniwersystet Jagiollonski w Krakowie

Regularity of solutions to the complex Monge-Ampere equation

In geometric analysis many arguments rely on a suitable regularity theory for the analyzed differential equations. Similarly to the solution of the Calabi conjecture often derving suitable a priori estimates is in fact the heart of the matter. In the talk a regularity result for the complex Monge-Ampe`re equation will be presented. We will prove that any C1,1 smooth plurisubharmonic solution u to the problem det(uij) = f with f strictly positive and H¨older continuous has in fact H¨older continuous second derivatives. For smoother f this follows form the classical Evans-Krylov theory yet in our case it cannot be applied directly. Instead we shall follow closely an idea of Xu-Jia Wang. Finally we shall discus how this particular regularity result can be used to justify an argument in the proof of uniqueness of metrics of constant scalar curvature by Chen and Tian.