Canonical metrics on Kahler manifolds and Monge-Ampere equations

The existence of canonical metrics on Kahler manifolds is a long standing open problem in complex geometry. It relies on the solvability of nonlinear partial differential equations of Monge-Ampere type. This project aims to establish the existence of canonical metrics by introducing new methods such as a variational approach to Abreu's equation and proving new Sobolev and Moser-Trudinger type inequalities in complex geometry. These new methods and results can also be applied to mechanics, string theory and mathematical physics. Success in this project will have a substantial influence on the interaction between geometry and differential equations.