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Index one minimal surfaces and the isoperimetric problem in spherical space forms

Dos Santos Viana, C; (2018) Index one minimal surfaces and the isoperimetric problem in spherical space forms. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

The research carried out in this thesis concerns two important class of stationary surfaces in Differential Geometry, namely isoperimetric surfaces and index one minimal surfaces. The former are solutions of the so called isoperimetric problem, which is to determine the regions of least perimeter among regions of same volume in a given manifold. The latter are critical points of the area functional with Morse index one, i.e., minimal surfaces which admits only one direction where the surface can be deformed so to decrease its area. These are usually constructed via mountain pass arguments. This work focus on the study of these objects when the ambient space is a 3-dimensional spherical space forms, i.e., space form with positive curvature. Our main results classify, at the level of topology, such stationary surfaces in the spherical space forms with large fundamental group. Our first result proves that the solutions of the isoperimetric problem in spherical space forms with large fundamental group are either spheres or tori. It was previously known that solutions with genus zero and one are respectively totally umbilical and flat. Combining our result and this geometric description, we derive that the solutions of the isoperimetric problem are either geodesic spheres or quotients of Clifford tori. Our second result proves that orientable minimal surfaces with index one in the aforementioned spherical space forms have genus at most two. This is a sharp estimate as one can use the continuous one-parameter min-max theory to construct in every 3-dimensional spherical space form an index one minimal surface with genus equal the Heegaard genus of such space which is known to be at most two. Our result confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Index one minimal surfaces and the isoperimetric problem in spherical space forms
Event: University College London
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Copyright © The Author 2018. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: https://discovery.ucl.ac.uk/id/eprint/10061744
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