{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86989"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86989","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Saddle Surfaces","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'.","abstract_html":"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'.","abstract_has_math":false,"creators":["Kalikakis, Dimitrios Emmanuel"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Igor G. Nikolaev"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:30Z","date_published":"2015-09-28T15:20:30Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9955633"],"render_values":[{"text":"(MiAaPQ)AAI9955633","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86989","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Igor G. 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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'.","Made available in DSpace on 2015-09-28T15:20:30Z (GMT). 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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'.","Made available in DSpace on 2015-09-28T15:20:30Z (GMT). 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