Classificação das superfícies mínimas de índices de morse pequeno imersas nos espaços S^3 e S^2 X S^1

Abstract

The purpose of this thesis is to discuss a conjecture classification concerning the index of non-totally geodesic minimal hypersurfaces of the n-dimensional standard sphere of radius one Sn. Briey discuss the basic theory of minimal submanifolds before focusing our attention to the minimal submanifolds and hypersurfaces of Sn. We present someresults of Simons which show that any minimal submanifolds of Sn is unstable, and how the totally geodesic Sk C Sn 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. Finally, we have the ultimate goal of studying the classification of minimal surfaces of Morse small index in product varieties S2 x S1(r).