Fully nonlinear geometric partial differential equations

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The primary goal of this project is to develop fundamental analytic tools for fully nonlinear equations airing from problems in differential geometry. The first project will build on the PI's work with Dinew and Zhang which proved the interior Holder continuity of the second order derivative for the weak solution of complex Monge-Ampere equations. This estimate is essential in studying the Kahler metric whose potential is of weak regularity. The PI proposes to establish this regularity estimate under more optimal condition. Along this direction, the PI will also study some interior estimates for the complex Monge-Ampere type equations. In the second project, the PI is determined to extend his recent work with Wang about the Alexandrov-Bakelman-Pucci (ABP) estimate on general Riemannian manifolds. The PI will investigate more interesting applications of this ABP method that is not broadly exploited in geometric analysis yet. It is possible to apply this technique to study the first proposed project on the complex Monge-Ampere equations. The third project concerns the mean curvature flows for Lagrangian submanifolds with boundaries which is related to a boundary value problem of a fully nonlinear equation. If the analytic parts were well understood, it would provide a powerful tool to construct special Lagrangian submanifolds with boundaries and to study the existence of the area-preserving minimal maps between bounded domains.This project aims at studying problems arising from differential geometry via fully nonlinear elliptic and parabolic equations. The goal is to better understand relations between geometric quantities and properties of these important geometric fully nonlinear equations. The innovative techniques developed in this proposal will lead to the solutions of important problems in geometry and enriching the existing theory of fully nonlinear partial differential equations in general. In the process, this work is having important consequences in both physics and applied sciences.