Columbia Geometry & Analysis Seminar

In this talk we will study the Morse index of minimal hypersurfaces and more generally, minimal submanifolds in projective spaces.

First we look at stable minimal submanifolds. We will talk about works of Simons, Lawson-Simons and Ohnita on the classification of compact stable minimal submanifolds in all compact rank one symmetric spaces, and consider the generalization to product manifolds. We will show that there do not exist odd-dimensional compact stable minimal immersions in the product of two complex projective spaces, and classify the stable minimal submanifolds in the product of a quaternionic projective space (resp. octonionic projective plane) with any other Riemannian manifold. This part is joint work with Alejandra Ramirez Luna (UCSC).

Next, we specialize to minimal hypersurfaces and consider higher Morse index. We will show that for an embedded unstable one-sided minimal hypersurface of the (n+1)-dimensional real projective space, the Morse index is at least n+2, and this bound can be attained by the cubic isoparametric hypersurfaces.