Special metrics in classical and generalized Kaehler geometry

这一建议涉及在复变数研究中自然产生的几何微分方程解的存在性、模和特殊的几何性质(类似于广义相对论中的爱因斯坦场方程)。其原型是紧复曲线上常高斯曲率度量的存在唯一性问题，它的分解导致了紧复曲线的著名的均匀化。在20世纪80年代，S，Calabi提出了一个更一般的问题，即在紧的Kaehler簇的上同调类中寻找规范的Kaehler度量，称为极值。从全局分析的角度出发，极值Kaehler度量由一类目前尚无通用方法的四阶非线性偏微分方程解给出。特殊情况包括常数量曲率的Kaehler度量和Kaehler-Einstein度量。

This proposal deals with the existence, moduli, and special geometrical properties of solutions of geometric differential equations (analogous to Einstein's field equation in general relativity) naturally arising in the study of complex varieties. The prototype is the problem of existence and uniqueness of a metric of constant Gauss curvature on a compact complex curve, whose resolution leads to the famous uniformization of compact complex curves. In the 1980's, Calabi formulated a more general problem of finding canonical Kaehler metrics, called extremal, in a given cohomology class of a compact Kaehler variety. From the point of view of global analysis, extremal Kaehler metrics are given by solutions to a non-linear 4th order PDE for which no general methods are currently available. Particular cases include Kaehler metrics of constant scalar curvature and Kaehler-Einstein metrics.