On the existence of harmonic mappings between doubly connected domains

TL;DR

This paper establishes conditions under which harmonic diffeomorphisms exist between doubly connected domains, extending the understanding beyond conformal mappings by relating domain thickness to harmonic map existence.

Contribution

It provides a precise criterion for the existence of harmonic diffeomorphisms based on the conformal moduli of doubly connected domains, filling a gap in the theory.

Findings

Existence of harmonic diffeomorphisms depends on the conformal moduli.

A threshold epsilon ensures harmonic maps exist for domains close in moduli.

The result applies to Dini-smooth doubly connected domains.

Abstract

While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain $\Omega^*$ there exists $\epsilon>0$ such that for every doubly connected domain $\Omega$ with $\operatorname{Mod} \Omega^*<\operatorname{Mod}\Omega<\operatorname{Mod} \Omega^*+\epsilon$ there exists a harmonic diffeomorphism from $\Omega$ onto $\Omega^*$.