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https://hdl.handle.net/2142/86989
Description
Title
Saddle Surfaces
Author(s)
Kalikakis, Dimitrios Emmanuel
Issue Date
2000
Doctoral Committee Chair(s)
Igor G. Nikolaev
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
The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that every energy minimizing surface in a nonpositively curved Aleksandrov's space is a saddle surface. Further, we show that the notion of a saddle surface is well defined for, a general Frechet surface and we prove that the space of saddle surfaces in an reals0 domain is complete in the Frechet distance. We also prove a compactness theorem for saddle surfaces in realskappa domains; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. Finally, we show that a saddle surface in a three-dimensional space of nonzero constant curvature kappa is a space of curvature not greater than kappa in the sense of A. D. Aleksandrov, which generalizes a classical theorem by S. Z. Shefel'.
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