Special metrics in classical and generalized Kaehler geometry

这一提议涉及几何微分方程（类似于广义相对论中的爱因斯坦场方程）解的存在性、模和特殊几何性质，这些解是在复杂多样性的研究中自然产生的。 原型问题是紧致复曲线上常高斯曲率度量的存在唯一性问题，其解决导致了著名的紧致复曲线的一致化。在20世纪80年代，卡拉比提出了一个更一般的问题，即在紧致凯勒簇的给定上同调类中找到规范的凯勒度量，称为极值。从全局分析的角度出发，给出了极值Kaehler度量的解的非线性四阶偏微分方程，目前还没有通用的方法。特殊情况包括常数标量曲率的凯勒度量和凯勒-爱因斯坦度量。

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.