Beltrami equation for the harmonic diffeomorphisms between surfaces
Differential Geometry
2020-07-15 v3 Complex Variables
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.
Cite
@article{arxiv.1903.05420,
title = {Beltrami equation for the harmonic diffeomorphisms between surfaces},
author = {Anestis Fotiadis and Costas Daskaloyannis},
journal= {arXiv preprint arXiv:1903.05420},
year = {2020}
}