Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane

TL;DR

This paper proves the existence of area-preserving minimal Lagrangian diffeomorphisms between convex domains of equal area in hyperbolic surfaces, advancing understanding of geometric mappings in negatively curved spaces.

Contribution

It establishes the existence of minimal Lagrangian diffeomorphisms between convex domains in hyperbolic surfaces, a novel result in differential geometry.

Findings

Existence of area-preserving minimal Lagrangian diffeomorphisms

Construction of such diffeomorphisms in hyperbolic geometry

Extension of minimal surface theory to convex domains

Abstract

Let N be a complete, simply-connected surface of constant curvature \kappa \leq 0. Moreover, suppose that \Omega and \tilde{\Omega} are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from \Omega to \tilde{\Omega} whose graph is a minimal submanifold of N \times N.