# Beltrami equation for the harmonic diffeomorphisms between surfaces

## TL;DR

This paper links harmonic diffeomorphisms between surfaces to a specific Beltrami equation, showing solutions relate to the sinh-Gordon equation, and classifies these maps based on this relationship.

## Contribution

It establishes a reduction of harmonic diffeomorphisms to a Beltrami equation framework and classifies solutions via the sinh-Gordon equation for constant curvature surfaces.

## Key findings

- Solutions for the Beltrami equation are derived for constant curvature surfaces.
- Harmonic diffeomorphisms are classified through sinh-Gordon equation solutions.
- The study provides a unified approach to solutions in the constant curvature case.

## Abstract

In this article it is shown that the study of harmonic diffeomorphisms, with nonvanishing Hopf differential, reduces to the study of the Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami function coincides with the imaginary part of the logarithm of the Hopf differential, therefore is a harmonic function. The real part of the logarithm of the Beltrami function satisfies an elliptic nonlinear differential equation, which in the case of constant curvature is an elliptic sinh-Gordon equation. Solutions are calculated for the constant curvature case in a unified way. The harmonic maps are therefore classified by the classification of the solutions of the sinh-Gordon equation.