AbstractsMathematics

On a Class of Complex Monge-Ampère Type Equations on Hermitian Manifolds

by Wei Sun




Institution: The Ohio State University
Department: Mathematics
Degree: PhD
Year: 2013
Keywords: Mathematics
Record ID: 2000758
Full text PDF: http://rave.ohiolink.edu/etdc/view?acc_num=osu1366286119


Abstract

We study the Dirichlet problem for a class of complex Monge-Ampère equations on Hermitian manifolds with smooth boundary and data, which turns to second order fully nonlinear elliptic differential equations. Under the condition that there exists an admissible subsolution, we solve the problem by method of continuity. To apply the standard arguments, the key step is to derive a priori estimates up to second order derivatives. Also, we are interested in the equations on closed manifolds, i.e. compact manifolds without boundary. A new relationship between ¿ and ¿ is discovered, and can help us derive sharper C<sup>2</sup> estimates. Based on the new estimates, we can derive the C<sup>0</sup> estimate and then C<sup>8</sup> estimates on closed Hermitian manifolds. Besides the method of continuity, parabolic flow method is also an effective way to solve second order elliptic equations. We introduce the related parabolic flows, and investigate the regularity and existence of solutions to the flows. To apply Evans-Krylov theory and Schauder estimates, we establish a priori estimates up to second order derivatives of admissible solutions. As a result, an admissible solution converges to a stationary solution, which solves the Dirichlet problem.