The Global Solutions to Cartan's Realization Problem

Abstract

We introduce a systematic method to solve a type of Cartan's realization problem. Our method builds upon a new theory of Lie algebroids and Lie groupoids with structure group and connection. This approach allows to find local as well as complete solutions, their symmetries, and to determine the moduli spaces of local and complete solutions. We apply our method to the problem of classification of extremal Kähler metrics on surfaces.

arXiv commentTypos corrected. Several comments added and a correction to an example in Section 9.2. This is the final version accepted for publication in Memoirs of the AMS