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ON ALMOST STABLE CMC HYPERSURFACES IN MANIFOLDS OF BOUNDED SECTIONAL CURVATURE

Part of: Global differential geometry

Published online by Cambridge University Press:  10 September 2019

JULIEN ROTH Open the ORCID record for JULIEN ROTH [Opens in a new window]  and
ABHITOSH UPADHYAY Open the ORCID record for ABHITOSH UPADHYAY [Opens in a new window]
JULIEN ROTH
Affiliation:
Laboratoire d’Analyse et de Mathématiques Appliquées, UPEM-UPEC, CNRS, F-77454 Marne-la-Vallée, France email [email protected]
ABHITOSH UPADHYAY*
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India email [email protected], [email protected]
*
✉[email protected], [email protected]
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Abstract

We show that almost stable constant mean curvature hypersurfaces contained in a sufficiently small ball of a manifold of bounded sectional curvature are close to geodesic spheres.

Keywords

stable constant mean curvaturehypersurfacegeodesic sphere

MSC classification

Primary: 53C24: Rigidity results
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 2 , April 2020 , pp. 333 - 338
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author gratefully acknowledges the support of a National Post-doctoral Fellowship (file no. PDF/2017/001165) from the Science and Engineering Research Board, Government of India.

References

Baikoussis, C. and Koufogiorgos, T., ‘The diameter of an immersed Riemannian manifold with bounded mean curvature’, J. Aust. Math. Soc. A 31 (1981), 189–192.Google Scholar
Barbosa, J. L. and do Carmo, M. P., ‘Stability of hypersurfaces with constant mean curvature’, Math. Z. 185(3) (1984), 339–353.Google Scholar
Barbosa, J. L., do Carmo, M. P. and Eschenburg, J., ‘Stability of hypersurfaces of constant mean curvature in Riemannian manifolds’, Math. Z. 197(1) (1988), 123–138.Google Scholar
Grosjean, J. F. and Roth, J., ‘Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces’, Math. Z. 271(1) (2012), 469–488.Google Scholar
Heintze, E., ‘Extrinsic upper bound for 𝜆1’, Math. Ann. 280 (1988), 389–402.Google Scholar
Hoffman, D. and Spruck, J., ‘Sobolev and isoperimetric inequalities for Riemannian submanifolds’, Comm. Pure Appl. Math. 27 (1974), 715–727.Google Scholar
Montiel, S. and Ros, A., ‘Compact hypersurfaces: the Alexandrov theorem for higher order mean curvature’, in: Differential Geometry, Pitman Monographs and Surveys in Pure and Applied Mathematics, 52 (eds. Lawson, B. and Tenenblat, K.) (Longman, Harlow, UK, 1991), 279–296.Google Scholar
Roth, J. and Scheuer, J., ‘Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space’, Ann. Global Anal. Geom. 51(3) (2017), 287–304.Google Scholar