@phdthesis{Kraus2003,
  author    = {Kraus, Daniela},
  title     = {Conformal pseudo-metrics and some applications},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-9193},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2003},
  abstract  = {The point of departure for the present work has been the following free boundary value problem for analytic functions \$f\$ which are defined on a domain \$G \subset \mathbb{C}\$ and map into the unit disk \$\mathbb{D}= \{z \in \mathbb{C} : |z|<1 \}\$. Problem 1: Let \$z_1, \ldots, z_n\$ be finitely many points in a bounded simply connected domain \$G \subset \mathbb{C}\$. Show that there exists a holomorphic function \$f:G \to \mathbb{D}\$ with critical points \$z_j\$ (counted with multiplicities) and no others such that \$\lim_{z \to \xi} \frac{|f'(z)|}{1-|f(z)|^2}=1\$ for all \$\xi \in \partial G\$. If \$G=\mathbb{D}\$, Problem 1 was solved by K?nau [5] in the case of one critical point, and for more than one critical point by Fournier and Ruscheweyh [3]. The method employed by K?nau, Fournier and Ruscheweyh easily extends to more general domains \$G\$, say bounded by a Dini-smooth Jordan curve, but does not work for arbitrary bounded simply connected domains. In this paper we present a new approach to Problem 1, which shows that this boundary value problem is not an isolated question in complex analysis, but is intimately connected to a number of basic open problems in conformal geometry and non-linear PDE. One of our results is a solution to Problem 1 for arbitrary simply connected domains. However, we shall see that our approach has also some other ramifications, for instance to a well-known problem due to Rellich and Wittich in PDE. Roughly speaking, this paper is broken down into two parts. In a first step we construct a conformal metric in a bounded regular domain \$G\subset \mathbb{C}\$ with prescribed non-positive Gaussian curvature \$k(z)\$ and prescribed singularities by solving the first boundary value problem for the Gaussian curvature equation \$\Delta u =-k(z) e^{2u}\$ in \$G\$ with prescribed singularities and continuous boundary data. This is related to the Berger-Nirenberg problem in Riemannian geometry, the question which functions on a surface R can arise as the Gaussian curvature of a Riemannian metric on R. The special case, where \$k(z)=-4\$ and the domain \$G\$ is bounded by finitely many analytic Jordan curves was treated by Heins [4]. In a second step we show every conformal pseudo-metric on a simply connected domain \$G\subseteq \mathbb{C}\$ with constant negative Gaussian curvature and isolated zeros of integer order is the pullback of the hyperbolic metric on \$\mathbb{D}\$ under an analytic map \$f:G \to \mathbb{D}\$. This extends a theorem of Liouville which deals with the case that the pseudo-metric has no zeros at all. These two steps together allow a complete solution of Problem 1. Contents: Chapter I contains the statement of the main results and connects them with some old and new problems in complex analysis, conformal geometry and PDE: the Uniformization Theorem for Riemann surfaces, the problem of Schwarz-Picard, the Berger-Nirenberg problem, Wittich's problem, etc.. Chapter II and III have preparatory character. In Chapter II we recall some basic results about ordinary differential equations in the complex plane. In our presentation we follow Laine [6], but we have reorganized the material and present a self-contained account of the basic features of Riccati, Schwarzian and second order differential equations. In Chapter III we discuss the first boundary value problem for the Poisson equation. We shall need to consider this problem in the most general situation, which does not seem to be covered in a satisfactory way in the existing literature, see [1,2]. In Chapter IV we turn to a discussion of conformal pseudo-metrics in planar domains. We focus on conformal metrics with prescribed singularities and prescribed non-positive Gaussian curvature. We shall establish the existence of such metrics, that is, we solve the corresponding Gaussian curvature equation by making use of the results of Chapter III. In Chapter V we show that every constantly curved pseudo-metric can be represented as the pullback of either the hyperbolic, the euclidean or the spherical metric under an analytic map. This is proved by using the results of Chapter II. Finally we give in Chapter VI some applications of our results. [1,2] Courant, H., Hilbert, D., Methoden der Mathematischen Physik, Erster/ Zweiter Band, Springer-Verlag, Berlin, 1931/1937. [3] Fournier, R., Ruscheweyh, St., Free boundary value problems for analytic functions in the closed unit disk, Proc. Amer. Math. Soc. (1999), 127 no. 11, 3287-3294. [4] Heins, M., On a class of conformal metrics, Nagoya Math. J. (1962), 21, 1-60. [5] K?nau, R., L?gentreue Randverzerrung bei analytischer Abbildung in hyperbolischer und sph?ischer Geometrie, Mitt. Math. Sem. Giessen (1997), 229, 45-53. [6] Laine, I., Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin - New York, 1993.},
  subject      = {Freies Randwertproblem},
  language  = {en}
}