Conformal welding

In mathematics, conformal welding (sewing or gluing) is a process in geometric function theory for producing a Riemann surface by joining together two Riemann surfaces, each with a disk removed, along their boundary circles. This problem can be reduced to that of finding univalent holomorphic maps f, g of the unit disk and its complement into the extended complex plane, both admitting continuous extensions to the closure of their domains, such that the images are complementary Jordan domains and such that on the unit circle they differ by a given quasisymmetric homeomorphism. Several proofs are known using a variety of techniques, including the Beltrami equation, the Hilbert transform on the circle and elementary approximation techniques.Sharon & Mumford (2006) describe the first two methods of conformal welding as well as providing numerical computations and applications to the analysis of shapes in the plane.

This method was first proposed by Pfluger (1960).

If f is a diffeomorphism of the circle, the Alexander extension gives a way of extending f to a diffeomorphism of the unit disk D:

where ψ is a smooth function with values in [0,1], equal to 0 near 0 and 1 near 1, and

with g(θ + 2π) = g(θ) + 2π.

The extension F can be continued to any larger disk |z| < R with R > 1. Accordingly in the unit disc

Now extend μ to a Beltrami coefficient on the whole of C by setting it equal to 0 for |z| ≥ 1. Let G be the corresponding solution of the Beltrami equation:

Let F₁(z) = G ∘ F⁻¹(z) for |z| ≤ 1 and F₂(z) = G (z) for |z| ≥ 1. Thus F₁ and F₂ are univalent holomorphic maps of |z| < 1 and |z| > 1 onto the inside and outside of a Jordan curve. They extend continuously to homeomorphisms f_i of the unit circle onto the Jordan curve on the boundary. By construction they satisfy the conformal welding condition:

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