Title: Harmonic Maps onto Polygonal Domains and Minimal Surfaces
Speaker: Professor Jane McDougall
Speaker Info: Colorado College
Brief Description:
Special Note:
Abstract:
A univalent harmonic map of the unit disk is a one to one complex valued harmonic function. The analytic dilatation is an analytic function that measures the degree to which the harmonic map is non-conformal. The Rado-Kneser-Choquet theorem guarantees the existence of univalent harmonic maps onto convex polygonal domains, but here I will give examples of harmonic maps onto non-convex domains, and show that for a given domain, a variety of dilatations in the form of finite Blaschke products are possible. In particular, a dilatation with an analytic square root allows for the existence of a third “height” function that, together with the harmonic map, parameterizes a minimal surface. Minimal surfaces over such domains were shown to exist in 1966 by Jenkins and Serrin via a non-constructive method. I will give examples of parameterizations of minimal surfaces over these domains, some of which replicate Karcher's "Jenkins Serrin surfaces" and provide new examples as well.Date: Thursday, January 19, 2006