Virtual immersions and minimal hypersurfaces in compact symmetric spaces
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Calculus of Variations
Virtual immersions and minimal hypersurfaces in compact symmetric spaces Ricardo A. E. Mendes1 · Marco Radeschi2 Received: 28 June 2019 / Accepted: 31 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Mathematics Subject Classification 49Q05 · 53A10 · 53C35
1 Introduction Let (M, g) be a Riemannian manifold, and a minimal immersed submanifold. This means that the second fundamental form of is traceless, or, equivalently, that is a critical point of the area functional. Then one is naturally led to consider variations up to second order, and to define the (Morse) index of as the dimension of the space of negative variations. When is closed, the index is finite. Many authors have developed methods to produce minimal submanifolds, including Min-Max Theory (see [6,14] for surveys), desingularization (see for example [8,13]), and equivariant methods (see for example [10–12]). For some of these the index of the minimal
Communicated by A. Neves. 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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Ricardo A. E. Mendes [email protected] Marco Radeschi [email protected]
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University of Oklahoma, Norman, USA
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University of Notre Dame, Notre Dame, USA 0123456789().: V,-vol
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submanifold is controlled, while for others the topology is controlled. On the other hand, the set of all minimal submanifolds of bounded index, area, or topology is the object of active research, in particular compactness results such as [4,7,17] have been obtained. Therefore it is natural to ask how the topology and the index of minimal submanifolds are related. One conjecture that fits in this framework is: (see [14, page 16], or [1, page 3] for a slightly different formulation) Conjecture (Marques–Neves–Schoen) Let (M, g) be a compact manifold with positive Ricci curvature, and dimension at least three. Then there exists C > 0 such that, for all closed embedded orientable minimal hypersurfaces → M, ind() ≥ Cb1 () where b1 () denotes the first Betti number of with real coefficients. Variations of this conjecture include replacing the assumption that the Ricci curvature is positive with other notions of positivity (or non-negativity) of the curvature; replaci
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