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Virtual immersions and minimal hypersurfaces in compact symmetric spaces

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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 results, 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.

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Acknowledgements

It is a pleasure to thank Lucas Ambrozio for many enlightening discussions during this project, especially regarding Lemma 14 and Theorem C(b). The final part of this project was carried out while the second-named author visited the University of Cologne. The second-named author wishes to thank Alexander Lytchak for his hospitality during the visit.

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Authors and Affiliations

  1. University of Oklahoma, Norman, USA

    Ricardo A. E. Mendes

  2. University of Notre Dame, Notre Dame, USA

    Marco Radeschi

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  1. Ricardo A. E. Mendes
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  2. Marco Radeschi
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Correspondence to Ricardo A. E. Mendes.

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Communicated by A. Neves.

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Ricardo A. E. Mendes received support from SFB 878: Groups, Geometry & Actions, DFG ME 4801/1-1 and NSF Grant DMS-2005373. Marco Radeschi received support from NSF Grant 1810913.

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Mendes, R.A.E., Radeschi, M. Virtual immersions and minimal hypersurfaces in compact symmetric spaces. Calc. Var. 59, 192 (2020). https://doi.org/10.1007/s00526-020-01854-x

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