About: 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. describe the first two methods of conformal weldin