Saddle Surfaces

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'.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Kalikakis, Dimitrios Emmanuel

Contributors dc:contributor

• Igor G. Nikolaev

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI9955633

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86989