Flat Projective Structures on Surfaces and Cubic Holomorphic Differentials

TL;DR

This paper explores the relationship between flat projective structures on surfaces, cubic holomorphic differentials, and elliptic PDEs, providing new insights into their geometric and analytical properties.

Contribution

It offers a novel interpretation of real projective structures via elliptic PDEs and applies these results to establish the uniqueness of minimal surfaces in symmetric spaces.

Findings

Interpretation of projective structures through elliptic PDEs

Connection between cohomology classes and geometric structures

Proof of uniqueness of minimal surfaces in symmetric spaces

Abstract

The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge theory. We shall also give an application of these results as the uniqueness of a minimal surface in a symmetric space.