Mathematical Sciences: Complex Geometry and Nonlinear Analysis

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The principal investigator will continue his research in complex differential geometry and nonlinear analysis. He will be particularly concerned with a study of Kahler manifolds through Kahler-Einstein metrics. In the 1950's Calabi asked if one could find a Kahler-Einstein metric on a compact Kahler manifold with positive first Chern class and no nontrivial holomorphic vector fields. Tian will try to answer this question by obtaining estimates for the associated Monge-Ampere operator on the underlying Kahler manifold. The methods involve nonlinear partial differential equations, compactness arguments, and collapsing of Riemannian manifolds. The existence of Kahler-Einstein metrics on spaces has become an important problem in differential geometry. It is related to the structure of space-time in which the curvature of space-time is proportional to the stress field. Spaces in which curvature is proportional to distance are called Einstein spaces. The principal investigator will try to determine if a class of spaces with a "complex" structure can be given the additional Einstein structure.