Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces

TL;DR

This paper proves that minimal Lagrangian diffeomorphisms between spherical cone surfaces are isometries, and shows that branched immersions of positively curved surfaces in Euclidean space are coverings of round spheres, extending classical rigidity results.

Contribution

It establishes a rigidity result for minimal Lagrangian diffeomorphisms on spherical cone surfaces without multiangle restrictions and generalizes Liebmann's theorem to branched immersions.

Findings

Minimal Lagrangian diffeomorphisms are isometries between spherical cone surfaces.

Branched immersions of positively curved surfaces are coverings of round spheres.

Extension of classical rigidity theorems to branched immersions.

Abstract

We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched immersion of a closed surface of constant positive Gaussian curvature in Euclidean three-space is a branched covering onto a round sphere, thus generalizing the classical rigidity theorem of Liebmann to branched immersions.