Boundary values versus dilatations of harmonic mappings |
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Authors: | D Bshouty W Hengartner |
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Institution: | (1) Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel;(2) Département des Mathématiques, Université Laval, G1K 7P4 Québec, Canada |
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Abstract: | This article is divided into two parts. In the first part, we consider univalent harmonic mappings from the unit diskU onto a Jordan domain Ω whose dilatation functions
have modulus one on an interval of the unit circle. The boundary values off depend very strongly on the values ofa(e
it
). A complete characterization of the inverse imagef
-1
(q) of a pointq on ∂Ω is given. We then consider the case where the dilatation functiona(z) is a finite Blaschke product of degreeN. It is shown that in this case, Ω can have at mostN+2 points of convexity. Finally, we give a complete characterization of simply connected Jordan domains Ω with the property
that there exists a nonparametric minimal surface over Ω such that the image of its Gaussian map is the upper half-sphere
covered exactly once. |
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Keywords: | |
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