Special hermitian metrics in complex geometry

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This proposal deals with the existence, moduli, and special properties of solutions of geometric differential equations (analogous to the Einstein 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 theorem for compact complex curves. In higher dimensions, the natural extension is the problem proposed by Calabi in the 1980's of finding canonical Kaehler metrics, called extremal, in a given cohomology class of a compact Kaehler variety. There are some other related problems considered in the proposal, which naturally arise in complex geometry, and one recurrent theme is a strong intertwining of algebraic geometry, differential geometry, and global analysis, with each providing complementary insight into the structure of the spaces of solutions.