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Compactness of the Space of Minimal Hypersurfaces with Bounded Volume and p-th Jacobi Eigenvalue

Abstract

Given a closed Riemannian manifold of dimension less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for \(p\ge 2\) one gains strong convergence to a smooth limit submanifold away from at most \(p-1\) points.

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Notes

  1. 1.

    Throughout this paper, we shall always tacitly assume all hypersurfaces to be connected.

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Acknowledgments

The authors wish to express their gratitude to Prof. André Neves for a number of enlightening conversations. During the preparation of this article, L.A. was supported by CNPq-Brazil, while A.C. and B.S. were supported by Prof. André Neves through his Leverhulme and European Research Council Start Grant.

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Correspondence to Alessandro Carlotto.

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Ambrozio, L., Carlotto, A. & Sharp, B. Compactness of the Space of Minimal Hypersurfaces with Bounded Volume and p-th Jacobi Eigenvalue. J Geom Anal 26, 2591–2601 (2016). https://doi.org/10.1007/s12220-015-9640-4

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Keywords

  • Minimal submanifolds
  • Compactness
  • Jacobi operator

Mathematics Subject Classification

  • 53A10
  • 49Q05