Extremal Conformal Structures on Projective Surfaces

Thomas Mettler

Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany

Date: December 18, 2018

Published Version: Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XX (2020), 1621—1663

1 Introduction
2 Projective and conformal structures
2.1 Preliminaries
2.2 Projective structures
2.3 Conformal structures
2.4 Compatibility of projective and conformal structures
2.5 A diffeomorphism invariant functional
3 Projective surfaces and associated bundles
3.1 Cartan's normal projective connection
3.2 The twistor space
3.3 An indefinite Kähler-Einstein 3-fold
3.4 The canonical flat case
4 The variational equations
4.1 The bundle of complex linear coframes
4.2 Submanifold theory in the twistor space
4.3 Derivation of the variational equations
5 Existence of critical points
5.1 Convex projective structures
5.2 Concluding remarks
Appendix I A Gauss--Bonnet type identity
References