Lect Minda Iwhmhm09[1]

Title: Conformal metrics in complex analysis Speaker: David Minda, University of Cincinnati Most of the material on which I will lecture is part of work with Alan Beardon of the University of Cambridge to be part of a book. The overall theme of the lectures is the role of conformal metrics in complex analysis. I plan to begin at a basic level and give largely expository lectures. Lecture 1: Conformal metrics and the three classical geometries. Conformal metrics play a fundamental role in complex analysis; they are special types of Riemannian metrics on plane regions that give angles their usual Euclidean measure. The three classical geometries, hyperbolic, Euclidean and spherical, arise from complete conformal metrics with curvature −1, 0 and +1. We will discuss these metrics and identify their isometry groups which are subgroups of the group of M¨obius transformations. We discuss why the hyperbolic metric is fundamental in geometric function theory. Lecture 2: The hyperbolic metric on simply connected regions. The hyperbolic metric on the unit disk D can be transplanted to any simply connected region that is conformally equivalent D by using a Riemann mapping of the region onto the unit disk. Covering theorems for univalent functions can be recast as comparisons between the hyperbolic metric and the quasihyperbolic metric on a simply connected region. Also, classical growth and distortion theorems for univalent functions have analogs in terms of the hyperbolic metric. Lecture 3: The equivalence of conformal metrics with constant curvature and holomorphic (meromorphic) functions. Loosely speaking, locally injective bounded holomorphic functions and conformal metrics with constant curvature −1 are the same. Precisely, on a hyperbolic simply connected plane region Ω there is a homeomorphic correspondence between the family of locally injective bounded holomorphic functions and the family of conformal metrics with constant curvature −1. The original version of this theorem is due to Liouville. His ‘proof’ does not meet the standards of today’s rigor, but probably could be made into a rigorous proof. We sketch a rigorous function-theoretic proof of Liouville’s Theorem using basic notions such as the connection and Schwarzian of a conformal metric. Lecture 4: The hyperbolic metric on hyperbolic planar regions. By making use of the Uniformization Theorem, the hyperbolic metric can be transported to any planar region whose complement in the plane contains at least two points.

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