The Bernstein Center Freiburg

Informal Bernstein Seminar
Susanne Höllbacher
Department of Mathematics and Complex Analysis
Julius-Maximilian University
Würzburg, Germany
Harmonic Welding - A Method for Numerical Conformal Welding

Thursday, April 9th, 2009
9:00 h sharp
Library, first floor
BCCN Building
Hansastr. 9a
The conformal welding operation is a well-known construction in geometric function theory and can also be seen as a part of conformal eld theory (CFT). Furthermore, it plays a prominent role in the construction of Teichmüller spaces of Riemann surfaces which are covered by the disc. Conformal welding can be described as a joining together of the two complementary domains D and C^D along their common boundary curve S1 using a homeomorphism of the unit circle. After conformal mapping of the welded surface one obtains a simple closed curve in the plane as the image of the unit circle which is associated with the homeomorphism. Therefore, via this operation every Jordan curve is represented by a certain homeomorphism of the unit circle. The conformal welding problem consists in establishing the welding curve from a given homeomorphism. An analysis of this correspondence is driven by the following questions: For which homeomorphism does the welding curve exist? In which way are the geometrical properties of the welding curve represented in the corresponding homeomorphism? E. Sharon and D. Mumford exploit this principle for the analysis of 2D shapes (simple closed C1-curves in the plane) and their similarities by studying the properties of the corresponding homeomorphisms. Equipping the space of all di eomorphisms of S1 with the Weil-Petersson (WP) norm allows the comparison between them (cf. [13]). All these considerations are based on the central Conformal Welding Theorem stating that in case of a quasisymmetric homeomorphism the existence of a corresponding curve is assured. The computation of the Jordan curve as the solution of the conformal welding problem actually translates into a numerical problem. In [13] two diff erent methods are presented. The theory of circle packing [18] provides a further possibility for the computation of the curve. In this work, another method, called harmonic welding, will be established to appoximately solve the conformal welding problem. In chapter 2, a graphic approach to the conformal welding operation will give an idea of the underlying mathematical problem. Theorem 2.8 cites the classical Conformal Welding Theorem as a central result. In chapter 3, the conformal welding problem will be transformed into the equivalent harmonic welding problem. Based on this formulation of the mathematical problem an integral equation will be derived in chapter 4. Its solution fi nally provides the real part of the solution to the conformal welding problem. An algorithm for the numerical computation of the solution of the integral equation is presented in chapter 5. Finally the results of the algorithm will be demonstrated by means of three examples of homeomorphisms.
The talk is open to the public. Guests are cordially invited!