Minimal Lagrangian connections on compact surfaces

TL;DR

This paper introduces and classifies minimal Lagrangian connections on compact surfaces, showing their unique correspondence with properly convex projective structures, thus linking differential geometry with geometric structures on surfaces.

Contribution

It defines minimal Lagrangian connections on surfaces and classifies all such connections on compact oriented surfaces with non-zero Euler characteristic, establishing a correspondence with convex projective structures.

Findings

Complete classification of minimal Lagrangian connections on certain surfaces.

Every properly convex projective surface corresponds uniquely to a minimal Lagrangian connection.

Abstract

We introduce the notion of a minimal Lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non-vanishing Euler characteristic. Combining our classification with results of Labourie and Loftin, we conclude that every properly convex projective surface arises from a unique minimal Lagrangian connection.