Rothe, Melanie (2024)
Geometry and Topology of Bipolar Minimal Surfaces in the 5-Sphere.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00028164
Ph.D. Thesis, Primary publication, Publisher's Version
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Item Type: | Ph.D. Thesis | ||||
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Type of entry: | Primary publication | ||||
Title: | Geometry and Topology of Bipolar Minimal Surfaces in the 5-Sphere | ||||
Language: | English | ||||
Referees: | Mäder-Baumdicker, Prof. Dr. Elena ; Francisco, Prof. Dr. Torralbo ; Wang, Prof. Dr. Guofang | ||||
Date: | 2 October 2024 | ||||
Place of Publication: | Darmstadt | ||||
Collation: | 147 Seiten | ||||
Date of oral examination: | 13 September 2024 | ||||
DOI: | 10.26083/tuprints-00028164 | ||||
Abstract: | In the theory of closed minimal surfaces in the n-dimensional sphere, geometric and topological properties are closely intertwined. A classical question is whether there exist (primarily embedded) examples of every topological type -- an issue, that particularly touches several other geometric variational problems. The current state of the art provides a rich theory and long list of examples for closed minimal surfaces in the 3-sphere. However, knowledge about representatives in the individual topological classes and higher codimensions remains sparse. To this end, the main focus of this thesis lies on a specific class of minimal surfaces in the 5-sphere, so-called bipolar surfaces, which arise from minimally immersed surfaces in the 3-sphere. On the one hand, we will topologically classify the bipolar minimal surfaces induced by two families among the prominent closed minimal surfaces in the 3-sphere that were constructed by H. Blaine Lawson in 1970. In that context, a notable phenomenon is that, regarding topology and embeddedness, bipolar surfaces can differ significantly from the original surfaces in the 3-sphere. On the other hand, we will consider bipolar surfaces as part of a more general class of minimal surfaces in the 5-sphere. First, this leads to a deeper understanding of their geometric data. Finally, this will in fact enable us to prove that, under certain conditions, locally any immersed surface of the aforementioned class is congruent to a bipolar surface. |
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Status: | Publisher's Version | ||||
URN: | urn:nbn:de:tuda-tuprints-281645 | ||||
Classification DDC: | 500 Science and mathematics > 510 Mathematics | ||||
Divisions: | 04 Department of Mathematics > Applied Geometry | ||||
Date Deposited: | 02 Oct 2024 12:01 | ||||
Last Modified: | 07 Oct 2024 11:27 | ||||
URI: | https://tuprints.ulb.tu-darmstadt.de/id/eprint/28164 | ||||
PPN: | 521848431 | ||||
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