This project focuses on the study of the extremal metrics in Kaehler geometry, the Calabi flow and related PDEs, such as the complex Monge-Ampere equation and an operator recently suggested by S. Donaldson. The stability conjecture in Kaehler geometry relates the existence of the extremal metrics to the stability of the underlying complex projective variety. The stability conjecture and related problems are fundamental problems in Kaehler geometry, which has tight relation with the partial differential equations, complex analysis and algebraic geometry. The precise understanding of the Calabi flow will lead to solving such a conjecture. Inspired by Perelman's success in Hamilton's problem to solve geometrization conjecture by Ricci flow, the PI proposes to study the Calabi flow to attack the existence of extremal metrics. The key issue for the Calabi flow is the long time existence, the asymptotic behavior and related problems. Based on the previous work of the PI and his collaborators, he plans to further study the Calabi flow on Kaehler surfaces, in particular on Del Pezzo surfaces and toric surfaces. The PI also plans to study the regularity problem of related PDEs, such as the complex Monge-Ampere equation, which is a fundamental equation in Kaehler geometry and is also closely related to the extremal metrics and the Calabi flow. In particular, the PI plans to study whether some regularity result, which has been proved for the real Monge-Ampere equation, holds or not for the complex Monge-Ampere equation. The study of S. Donaldson's operator will give new insight on this. Donaldson's operator can be viewed as an operator with some nature of both the real and complex Monge-Ampere operators.
Problems in the proposal arise naturally from our attempts to understand geometric partial dierential equations from geometry and physics, which have tight relation with many other fields such as algebraic geometry, complex analysis and mathematical physics. One of the key features of these problems is how the global structure of a space influences the local, analytic properties of the solutions of such equations. Understanding this general principle will have broad impact in physics and other fields of science in general. The proposed research will have immediate beneficial effect on students in PI's home university.
