Special metrics in complex geometry and applications

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The PI proposes to study special metrics on complex manifolds and applications. These metrics are governed by non-linear partial differential equations, including and generalizing Einstein's equation for gravity. The problems raised in this proposal center around many active areas of mathematics, including differential geometry, algebraic geometry, partial differential equations, several complex variables, and mathematical physics. This interdisciplinary nature will lead to further interactions between the PI and experts in other areas. Also the topic of this proposal will be of interest to many graduate students in Stony Brook working in closely-related fields, and will facilitate the broadening training of both undergraduate and graduate students. After S.-T. Yau's renowned work on the Calabi conjecture, there has been extensive study of special metrics in Kahler geometry during the past few decades. The guiding conjecture is a correspondence between the existence of extremal Kahler metrics and certain stability notion arising from geometric invariant theory. Recent joint work of the PI with X.X. Chen and Sir Simon Donaldson proved the conjecture in the case of Kahler-Einstein metrics on Fano manifolds. The proposed research will take this result as a starting point and is intended to push the theory further. Firstly the PI would like to investigate the interplay between Kahler-Einstein metrics, moduli in algebraic geometry, and the AdS/CFT correspondence in physics; secondly he would like to study a geometric evolution equation, namely the Calabi flow, on a general polarized Kahler manifold, and investigate its connection with degenerations in algebraic geometry and geometry of the space of Kahler metrics.