The PI plans to continue research in three inter-related areas. The first concerns geometric approximation of Kaehler metrics by Bergman metrics. The work of Tian, Yau and Zelditch shows that Kaehler metrics can be smoothly approximated by Bergman metrics, and this ``quantization process has found numerous striking applications to Kaehler geometry. The PI, in joint work with Phong, has shown that this approximation holds as well at the geometric level: thus, geodesic segments and geodesic rays can be canonically approximated by segments and rays in the space of Bergman metrics. The first project is aimed at deepening our understanding of the structure of the space of Kaehler metrics: The PI will investigate higher dimensional analogues of such Bergman approximations using methods of PDE and pluri-potential theory. The second project concerns K-stability and constant scalar curvature metrics. Here the motivating question is a conjecture of Donaldson which asserts that the non-existence of a csc metric should imply the existence of a geodesic ray along which the K-energy decreases. The PI will study this via the geodesic rays associated to test configurations (constructed in joint work with Phong). The next key step in this program is to prove C^{1,1} regularity of these rays using some of the recent advances in pluri-potential theory. The third project concerns the Kaehler-Ricci flow. The goal is to relate the S-condition and the B-condition (introduced in joint work with Phong, and studied as well in joint work with Song and Weinkove) to some of the more classical notions of stability in algebraic geometry.
The main theme of this proposal concerns the Einstein equation, which originally arose in the theory of general relativity. The great insight from physics is that many of the mysteries of the universe can be explained if one accepts the notion that our universe is curved. This conceptual breakthrough may be compared to the discovery, dating back to the early Greek philosophers, that the surface of the earth is curved. The Einstein equation gives a mathematical formulation of the curvature properties of the universe which is precise enough to make accurate predictions for a vast range of large scale physical phenomena. It turns out, for reasons that remain quite obscure, that the same Einstein equations can also be used to resolve deep and longstanding problems in topology and geometry. The partial differential equations that arise in this approach are non-linear, and their study requires a broad range of tools from real and complex analysis, as well as algebraic and differential geometry. This project will investigate the interplay between these various branches of mathematics, with the goal of furthering the understanding of fundamental geometric structures.
