University of Illinois Urbana-Champaign

A New Family of Symmetric Constant Mean Curvature Surfaces

Groisman, Pavel P.

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Description

Title
A New Family of Symmetric Constant Mean Curvature Surfaces
Author(s)
Groisman, Pavel P.
Issue Date
2004
Doctoral Committee Chair(s)
John Sullivan
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
In this work, the author studies genus zero constant mean curvature (CMC) surfaces, Alexandrov-embedded in R3. In the first chapters, the work of Lawson, and also of Pinkall and Polthier is extended and the first-order description of correspondence is obtained between constant mean curvature surfaces in various space forms. As important example, the correspondence between Delaunay surfaces of revolution (unduloids and nodoids), and spherical helicoids is studied. In the second part, the author describes a family of CMC surfaces with n horizontal and 2 vertical ends. This family has the symmetry of the group Dn x Z2 and is called n-wheels. Based on the methods from the early chapters of the thesis and also on the Hopf projection, the candidate classification map is devised. The remaining chapters take many steps to show this map is a homeomorphism. It is shown that, given that the fundamental piece of the surface is a graph over a vertical plane, the map is a homeomorphism. The developed way of thinking can very likely be adapted to other families of symmetric CMC surfaces in R3. In each case, one would need to show the fundamental piece of the surface is a graph over a plane.
Graduation Semester
2004
Type of Resource
text
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http://hdl.handle.net/2142/86848

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