Rothe, Melanie (2024)
Geometry and Topology of Bipolar Minimal Surfaces in the 5-Sphere.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00028164
Dissertation, Erstveröffentlichung, Verlagsversion

Kurzbeschreibung (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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