Virtual immersions and minimal hypersurfaces in compact symmetric spaces

TL;DR

This paper establishes lower bounds on the index and nullity of minimal hypersurfaces in compact symmetric spaces, linking these bounds to topological invariants and introducing a generalized isometric immersion framework.

Contribution

It introduces a new framework of generalized isometric immersions with skew-symmetric second fundamental form for analyzing minimal hypersurfaces in symmetric spaces.

Findings

Lower bounds on index plus nullity depend linearly on the first Betti number.

Under certain conditions, the index alone has a lower bound affine in the first Betti number.

Characterization of compact symmetric spaces via skew-symmetric second fundamental form.

Abstract

We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a certain genericity condition, or if the ambient space is a product of two CROSSes, we improve this to a lower bound on the index alone, which is affine in the first Betti number. To prove these, we introduce a generalization of isometric immersions in Euclidean space. Compact symmetric spaces admit (and in fact are characterized by) such a structure with skew-symmetric second fundamental form.