Date of Award

8-26-2022

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Leonid Kovalev

Second Advisor

Joseph Paulsen

Keywords

complex derivative, conformal, global bound, nearly round

Subject Categories

Physical Sciences and Mathematics

Abstract

Geometric properties of a domain in the complex plane reflect important informationabout the conformal maps to and from the domain. We examine a variety of geometric properties and use them to construct explicit global distortion bounds for both the compression and stretching of conformal map. Compressive distortion is controlled when the modulus of the derivative of a complex function is bounded from below, expansive distortion when it is bounded above. For the initial set of results, we quantify the degree to which a convex domain is nearly round with two parameters; radii of the largest inscribed disk and smallest circumscribed disk. A third parameter captures information about curvature on the boundary. The three parameters are used to construct a global stretching bound for a conformal map of the unit disk onto the domain, or equivalently, a bound on compression in the other direction. The Möbius invariant Kulkarni-Pinkall metric is used in constructing these explicit bounds. Next we generalize the previous results by weakening the assumption of convexity to something slightly stronger than star-shaped. The parameter giving the radius of the largest inscribed disk is replaced a more relevant radius, that of the largest disk from which every point of the domain can be seen. Finally we turn to the bounds in the opposite direction, that is, stretching bounds on conformal maps from a convex domain onto the unit disk, and compression bounds from the disk onto a convex domain. We use the same two parameters to quantify the degree to which a domain is nearly round, but have no need of a curvature parameter in this case. The bound in this final chapter is shown to be the best possible.

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Open Access

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