# Conformal welding of rectifiable curves

In classical conformal welding theory, we start with a homeomorphism $$h$$ of the unit circle and try to find a Jordan domain $$D$$ together with two conformal isomorphisms $$f_1 \colon \mathbb D \to D$$ and $$f_2 \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \overline{D}$$. such that $$h|_{S^1} = f_1^{-1} \circ f_2$$. The existence of these two maps are guaranteed if $$h$$ is a quasisymmetric homeomorphism, and the uniqueness follows from uniqueness of complex dilatation.

My question is: Let $$h$$ be a homeomorphism of the unit circle, and let $$J(z) = z+1/z$$. Under what conditions can we find a rectifiable curve $$\Gamma$$ of length $$4$$ together with a conformal isomorphism $$f \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$$ such that $$f|_{S^1} = \gamma \circ J \circ h$$, where $$\gamma \colon [-2,2] \to \mathbb C$$ is an arc-length parameterization of $$\Gamma$$? What about uniqueness?

I would really appreciate any reference of previous work in this direction, as well as ideas and suggestions.

Some motivations for this question:

Right now, I am trying to classify all smooth curves (and eventually rectifiable curves or Jordan arcs) $$\Gamma$$ with a certain property that reduces to $$\gamma'(t) = \Phi_1^{-1}(\gamma(t)) \Phi_2^{-1}(\gamma(t)).$$ To explain the symbols, $$\gamma$$ is an arc-length parameterization of $$\Gamma$$, and $$\Phi_1^{-1}(z), \Phi_2^{-1}(z)$$ represent the two pre-images of $$z \in \Gamma$$ under the conformal isomorphism $$\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$$. These pre-images are distinct if $$z$$ is not one of the endpoints of $$\Gamma$$.

I know curves of constant curvature satisfy this property, and I have some reasons to believe that they are the only ones. I would hope the "uniqueness" of conformal welding of curves, in some appropriate sense, would help me with the classification.

• You stated the welding theorem incorrectly. The conformal isomorphisms are from the disk to SOME Jordan region and from the complement of the disk to the complement of this Jordan region.(Condormal isomorphisms of a disk onto itself are trivial: they are fractional-linear). Apr 15 '20 at 12:42
• @AlexandreEremenko That was a careless mistake on me. Already corrected! Apr 15 '20 at 16:46
• Could you add some motivation as to where your question comes from? Just curious... Apr 15 '20 at 17:00
• @MalikYounsi Happy to! Apr 15 '20 at 17:22