Einstein manifolds are geometric objects important in both mathematics and physics. In physics, they are used to describe the space-time in Einstein's theory of general relativity. In mathematics, they are basic building blocks of more complicated geometries. The study of Einstein manifolds is thus a basic problem in geometry. One effective way to construct Einstein manifolds is to require that the underlying manifold has a complex algebraic structure. In other words, the points of such a manifold are complex-valued solutions of polynomial equations. Einstein metrics on such algebraic manifolds are called Kaehler-Einstein metrics. In the late 70s, Aubin and Yau constructed Kaehler-Einstein metrics with negative Ricci curvatures. Yau also constructed Kaehler-Einstein metrics with zero Ricci curvatures, which are now called Calabi-Yau metrics and play important roles in the string theory of physics. On the other hand, only recently has people pinned down a sufficient and necessary condition, called K-stability, for the existence of Kaehler-Einstein metrics with positive Ricci curvatures for a class of algebraic manifolds called Fano manifolds. This result depends on the work of many people, most importantly by Tian and Donaldson. After these discoveries, we want to further our understandings of such Kaehler-Einstein metrics and the obstructions to their existence. These problems are the main concerns of the proposal. The study of these Kaehler-Einstein metrics will greatly improve our understanding of Einstein manifolds important in both physics and mathematics.
In this proposal, the PI will study the following closely related problems. 1.Various continuity methods of partial differential equations are used to solve the Kaehler-Einstein equation. The recent breakthroughs give qualitative pictures of blow up behaviors and convergences of these continuity methods. However, deeper quantitative understandings of the blow up behaviors or singularity forming phenomena are needed. The PI has studied in detail such quantitative properties for toric Fano manifolds. The PI will study the singularities forming processes for a broader class of Fano manifolds. The PI will also study the classification of the singularities formed in low dimensions by combining the methods from Riemannian geometry and algebraic geometry. 2.The PI will study concrete constructions of Kaehler-Einstein metrics and related canonical metrics. On the one hand, the PI likes to extend the construction of toric Kaehler-Einstein metrics to other Kaehler-Einstein metrics with large symmetries, for example, on spherical varieties. On the other hand, the PI will study the classification of Sasaki-Einstein metrics with large symmetries in low dimensions based on his calculations of important examples. Related methods will also be applied to construct Kaehler-Ricci solitons and extremal Kaehler metrics. 3.The PI will study the deformations of canonical Kaehler metrics including Kaehler-Einstein metrics and Kaehler-Ricci solitons, and to understand the moduli spaces of these canonical Kaehler metrics. He will also study the singularities on the boundaries of these moduli spaces. 4.The PI and his collaborator will study the K-stability using algebraic geometry based on their previous work on K-stability. They will use tools from minimal model program to test K-stability. This will allow us to get Kaehler-Einstein metrics using algebro-geometric methods.