Mathematical Sciences: Complex Geometry and Nonlinear Analysis

首席研究员将继续他在复杂微分几何和非线性分析方面的研究。他将特别关注通过卡勒-爱因斯坦度量学研究卡勒流形。上世纪50年代，S·卡拉比提出，是否可以在紧致Kahler流形上找到第一类为正且没有非平凡全纯向量场的Kahler-Einstein度规。为了回答这个问题，田将尝试通过获得基础Kahler流形上相关的Monge-Ampere算子的估计来回答这个问题。这些方法涉及到非线性偏微分方程组、紧性论证和黎曼流形的折叠。空间上的Kahler-Einstein度量的存在性已成为微分几何中的一个重要问题。它与时空的结构有关，其中时空的曲率与应力场成正比。曲率与距离成正比的空间称为爱因斯坦空间。首席研究人员将试图确定一类具有“复杂”结构的空间是否可以被赋予额外的爱因斯坦结构。

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.