  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
$\Omega^*$.
