PDE Seminar

Title: Harmonic Maps onto Polygonal Domains and Minimal Surfaces
Speaker: Professor Jane McDougall
Speaker Info: Colorado College
Brief Description:
Special Note:

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
Time: 3:00pm
Where: Lunt 105
Contact Person: Prof. Gui-Qiang Chen, Keith Burns
Contact email: gqchen@math.northwestern.edu
Contact Phone: 847-491-5553
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