# Lipschitz property of minimisers between double connected surfaces

## TL;DR

This paper investigates the Lipschitz regularity of energy-minimizing diffeomorphisms between doubly connected Riemann surfaces, revealing their harmonic map structure and special Hopf differentials.

## Contribution

It extends the understanding of minimizers' regularity to Riemann surfaces with smooth boundaries, building on harmonic map theory and Noether's principle.

## Key findings

- Minimizers are Noether harmonic maps with special Hopf differentials.
- The Lipschitz property of minimizers is established.
- The proof relies on deep geometric and analytical results from Jost and others.

## Abstract

We study the global Lipschitz character of minimisers of the Dirichlet energy of diffeomorphisms between doubly connected domains with smooth boundaries from Riemann surfaces. The key point of the proof is the fact that minimisers are certain Noether harmonic maps, with Hopf differential of special form, a fact invented by Iwaniec, Koh, Kovalev and Onninen in \cite{inv} for Euclidean metric and by the author in \cite{calculus} for the arbitrary metric, which depends deeply on a result of Jost \cite{Job1}.