Minimal diffeomorphism between hyperbolic surfaces with cone singularities

TL;DR

This paper proves the existence and uniqueness of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces with different cone angles less than π, expanding understanding of geometric mappings.

Contribution

It establishes the existence and uniqueness of minimal diffeomorphisms between hyperbolic cone surfaces with varying cone angles, a novel result in geometric analysis.

Findings

Existence of minimal diffeomorphism for cone angles < π

Uniqueness when cone angles of the first surface are smaller

Extension of minimal diffeomorphism theory to cone surfaces

Abstract

We prove the existence of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces $(\Sigma,g_1)$ and $(\Sigma,g_2)$ when the cone angles of $g_1$ and $g_2$ are different and smaller than $\pi$. When the cone angles of $g_1$ are strictly smaller than the ones of $g_2$, this minimal diffeomorphism is unique.