Online conference

Index and stability of compact minimal submanifolds in the Berger spheres

This 4-part lecture was part of the special session about constant mean curvature surfaces organized by José M. Manzano and Marcos Cavalcante in the conference Geometric Analysis: Past, present and future host online by Hojoo Lee.

The purpose of this youtube lecture is to show my joint work with Prof. F. Urbano about the “Index and stability of compact minimal submanifolds in the Berger spheres”.

Instead of starting directly talking about the Bergers spheres the first three parts presents the study of the stability of minimal submanifolds in the standard round sphere. We show that the sphere does not admit compact stable minimal submanifolds and we characterize those compact minimal submanifolds which fail short from being stable (the lowest index examples), a result due to Simons in the sixties.

More precisely:

The first part introduces the notion of minimal submanifolds through the computation of the critical points of the volumen functional. The second variation will provide with the notion of index and stability. Those who are familiar with these can skip Part I and go directly to Part II where some motivational examples will be given.

Part III discusses the classic result of Simons about the characterization of the great spheres as the low index minimal examples in the round sphere. Simons also gave a similar characterization by the nullity that will not be covered.

Finally, Part IV studies the same problem in the Berger spheres, which are nothing but a sphere endowed with a Riemannian metric obtained from the standard one by changing the length of the Hopf fibers.