Riemann-Hilbert problems, circle packing and conformal geometry
Final Report Abstract
In this project we have studied nonlinear boundary value problems for holomorphic functions with an emphasis on the emerging field of discrete function theory (circle packing) and applications in conformal differential geometry. We have established several extensions of the celebrated Beurling-Riemann mapping theorem, e.g. for the case of finitely many prescribed critical points and for the case of target domains other than the entire complex plane. We have also obtained a discrete version of Beurling's result for circle packings with prescribed branch structure. By solving a special case of the Berger-Nirenberg problem in differential geometry we have found a characterization of the critical sets of bounded holomorphic functions in terms of the zero sets of the weighted Bergman space A1^2. This has led to a number of new connections between complex analysis and differential geometry. For instance, we have proved that for every nonconstant bounded analytic function in the unit disk there is always a Blaschke product with the same critical points. This can be seen as an analogue of a classical result due to Riesz, but for the critical points instead of the zeros. These Blaschke products are obtained as the developing maps of maximal conformal Riemannian metrics and they form an interesting new class of Blaschke products, which we call maximal Blaschke products. Maximal Blaschke products do have a number of intriguing properties. For instance, we have established an extension of the well-known Schwarz-Nehari Lemma from 1949, we have shown that maximal Blaschke products form a semigroup w.r.t. to composition, and we have studied the boundary behaviour of maximal Blaschke products. These results strongly indicate that maximal Blaschke products provide faithful infinite analogues of finite Blaschke products and might be seen as serious candidates for "infinite branched coverings" of the unit disk. As a byproduct of our investigations, we have found that the Poincaré metric is a strongly submultiplicative domain function - a new and perhaps unexpected property of the Poincare metric for plane domains. In this way we have been led to define a new type of capacity for compact subsets of the complex plane. This so-called Poincaré capacity provides a generally much finer method to measure the size of a compact set than the standard logarithmic capacity. In the algorithmic and numerical parts of the project, we have developed algorithms and related software for solving various (boundary value) problems in circle packing in a unified setting. This includes conformal mapping, Riemann-Hilbert problems, the generalized Beurling problem, complex differential equations and various types ofthe Hilbert transform. The software can be conveniently operated by graphical user interfaces; direct interaction with CIRCLEPACK adds more features for generating, manipulating and displaying circle packings. Being able to solve these boundary value problems, opens up new ways of defining, creating and controlling faithful analogues of holomorphic functions in circle packing - in accordance with Bernhard Riemann's general concept of "understanding analytic functions from conditions imposed on their values at the boundary of the domain".
Publications
- Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky, Math. Z. (2011) 267 No. 3-4, 851-868
D. Kraus, O. Roth and T. Sugawa
- Circle packings as differentiable manifolds. Contrib. Algebra and Geometry (2012) 53 No 2, 399-420
D. Bauer, K. Stephenson, E. Wegert
- Critical points, the Gauss curvature equation and Blaschke products. Fields Institute Comm. (2012) 65, 133-157
D. Kraus and O. Roth
- On Beurling's boundary value problem in Circle Packing, Compl. Var. Elliptic Eq. (2012) 57 No. 2-4, 397-410
D. Kraus, O. Roth and E. Wegert
(See online at https://doi.org/10.1080/17476933.2011.598931) - Visual Complex Functions - An Introduction with Phase Portraits. Springer Basel 2012. 360p. 231 illus., 202 in color
E. Wegert
- Composition and decomposition of indestructible Blaschke products, Comput. Methods Panel. Theory (2013) 13 No. 2, 253-262
D. Kraus and O. Roth
(See online at https://doi.org/10.1007/s40315-013-0022-2) - Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature, Proc. London Math. Soc. (2013) 106 No. 4, 931-956
D. Kraus
(See online at https://doi.org/10.1112/plms/pds054) - Maximal Blaschke products, Adv. Math. (2013) 241, 58-78
D. Kraus and O. Roth
- Computing the Hilbert transform in wavelet bases on adaptive grids. Operator Theory: Advances and Applications (2014) 236, 337-356
F. Martin, E. Wegert
(See online at https://doi.org/10.1007/978-3-0348-0648-0_21)
