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Minimal hypersurfaces of the round sphere

University of British Columbia

The purpose of this thesis is to discuss a conjectured classification concerning the index of non-totally geodesic minimal hypersurfaces of the n-dimensional standard sphere of radius one S^n. We briefly discuss the basic theory of minimal submanifolds before turning our attention to minimal submanifolds and hypersurfaces in S^n. We present some results of Simons which show that any minimal submanifold of S^n is unstable, and how the totally geodesic S^k ⊂ S^n are characterized by their index. We then present a related conjecture which claims that the Clifford hypersurfaces are also characterized by their index in a similar way, discuss the most recent developments related to the conjecture, and give Urbano’s proof of the conjecture for the special case when n = 3

Text

2013-08-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/44774

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/