Virtual immersions and minimal hypersurfaces in compact symmetric spaces

2017/08/19 by Mendes, Ricardo A. E., Radeschi, Marco · 1 citation
#49Q05 #53A10 #53C35 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1708.05881

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.

Cited by

• Ricci curvature and minimal hypersurfaces with large Betti numbers

Related

• Robust index bounds for minimal hypersurfaces of isoparametric submanifolds and symmetric spaces
• Index estimate by first Betti number of minimal hypersurfaces in compact symmetric spaces
• Virtual immersions, and a characterization of symmetric spaces
• Index and first Betti number of f-minimal hypersurfaces: general ambients
• Comparing the Morse index and the first Betti number of minimal hypersurfaces