Extremal Problems for Curves in Metric Spaces of Curvature Bounded Above
Maneesawarng, Chaiwat
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https://hdl.handle.net/2142/86991
Description
Title
Extremal Problems for Curves in Metric Spaces of Curvature Bounded Above
Author(s)
Maneesawarng, Chaiwat
Issue Date
2000
Doctoral Committee Chair(s)
Stephanie B. Alexander
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
We consider extremal problems involving curvature and curve length in metric spaces of curvature bounded above in the sense of Alexandrov. To a large extent, these problems have previously been solved only in Euclidean space. In addition to a nontrivial extention of the definition of total curvature, we give here sharp estimates of the length of a curve, one in terms of total curvature and chordlength, and another in terms of total curvature and the radius of a circumball. The latter gives rise to extremal configurations that have not previously been seen. We also establish a comparison theorem for the chordlength of a curve whose geodesic curvature is bounded by a function.
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