Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.763318
Title: Index one minimal surfaces and the isoperimetric problem in spherical space forms
Author: Dos Santos Viana, C.
ISNI:       0000 0004 7661 2230
Awarding Body: UCL (University College London)
Current Institution: University College London (University of London)
Date of Award: 2018
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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.
Supervisor: Neves, A. ; Lotay, J. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.763318  DOI: Not available
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