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Limit Domains in Several Complex Variables
Console, Alexander
Console, Alexander
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Abstract
In this thesis we demonstrated the existence of domains in C^2 evidencing both intrinsic phenomena of C^n, n 1, and different types of boundary smoothness. We constructed these domains by taking limits of preimages of polydiscs under a sequence of shears selected to control boundary smoothness. Unlike the complex plane, in C^n there are simply connected domains that are biholomorphic to C^n but are proper subsets of C^n. These domains are called FatouBieberbach domains and they arise naturally in the study of complex dynamics. We showed that there exists a Fatou-Bieberbach domain in C^2 with Gevrey smooth boundary. Another interesting occurrence in C^2 is the existence of simply connected proper subsets of C^2 that are not biholomorphic to the unit ball nor biholomorphic to C^2. Onesuch class are Short-C^2 domains. We constructed Short-C^2 domains with C-infinity boundary and Short-C^2 domains with prescribed local C^l boundary smoothness and controlled geometry.
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Date
2013-12-31
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University of Kansas
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This item contains archived web content.
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Keywords
Mathematics, Complex, Variables