Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T12:42:34.275Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEW RESULT ABOUT ALMOST UMBILICAL HYPERSURFACES OF REAL SPACE FORMS

Part of: Global differential geometry

Published online by Cambridge University Press:  14 October 2014

JULIEN ROTH
JULIEN ROTH*
Affiliation:
Laboratoire d’Analyse et de Mathématiques Appliquées, UPEM-UPEC, CNRS, F-77454 Marne-la-Vallée, France email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this short note, we prove that an almost umbilical compact hypersurface of a real space form with almost Codazzi umbilicity tensor is embedded, diffeomorphic and quasi-isometric to a round sphere. Then, we derive a new characterisation of geodesic spheres in space forms.

Keywords

almost umbilical hypersurfacesspace formsconstant scalar curvature

MSC classification

Secondary: 53C20: Global Riemannian geometry, including pinching 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 1 , February 2015 , pp. 145 - 154
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Alencar, H., Do Carmo, M. P. and Rosenberg, H., ‘On the first eigenvalue of the linearized operator of the rth mean curvature of a hypersurface’, Ann. Global Anal. Geom. 11 (1993), 387–395.CrossRefGoogle Scholar
Alexandrov, A. D., ‘A characteristic property of spheres’, Ann. Mat. Pura Appl. (4) 58 (1962), 303–315.Google Scholar
Cheng, Q. M., ‘Complete hypersurfaces in a Euclidean space ℝn+1 with constant scalar curvature’, Indiana Univ. Math. J. 51 (2002), 53–68.CrossRefGoogle Scholar
Cheng, S. Y. and Yau, S. T., ‘Hypersurfaces with constant scalar curvature’, Math. Ann. 225 (1977), 195–204.CrossRefGoogle Scholar
Colbois, B. and Grosjean, J. F., ‘A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space’, Comment. Math. Helv. 82 (2007), 175–195.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, D., ‘Sobolev and isoperimetric inequalities for Riemannian submanifolds’, Comm. Pure Appl. Math. 27 (1974), 715–727.Google Scholar
Li, H., ‘Hypersurfaces with constant mean curvature in space forms’, Math. Ann. 305 (1996), 665–672.Google Scholar
Michael, J. H. and Simon, L. M., ‘Sobolev and mean-value inequalities on generalized submanifolds of R n’, Comm. Pure Appl. Math. 26 (1973), 361–379.Google Scholar
Okayasu, T., ‘On compact hypersurfaces in a Euclidean space with constant scalar curvature’, Kodai Math. J. 28 (2005), 577–585.Google Scholar
Ros, A., ‘Compact hypersurfaces with constant higher order mean curvatures’, Rev. Mat. Iberoam. 3 (1987), 447–453.CrossRefGoogle Scholar
Ros, A., ‘Compact hypersurfaces with constant scalar curvature and a congruence theorem’, J. Differential Geom. 27 (1988), 215–220.Google Scholar
Roth, J., ‘Rigidity results for geodesic sphere in space forms’, in: Differential Geometry, Proc. VIII Int. Colloq. Differential Geometry, Santiago de Compostela (World Scientific, Singapore, 2009), 156–163.Google Scholar
Roth, J., ‘Une nouvelle caractérisation des sphères géodésiques dans les espaces modèles’, C. R. Math. 347(19–20) (2009), 1197–1200.Google Scholar
Roth, J., ‘A remark on almost umbilical hypersurfaces’, Arch. Math. (Brno) 49(1) (2013), 1–7.Google Scholar
Scheuer, J., ‘Quantitative oscillation estimates for almost-umbilical closed hypersurfaces in Euclidean space’, 2014, arXiv:1404.2525.CrossRefGoogle Scholar
Shiohama, K. and Xu, H., ‘Rigidity and sphere theorems for submanifolds’, Kyushu J. Math. 48(2) (1994), 291–306.Google Scholar
Shiohama, K. and Xu, H., ‘Rigidity and sphere theorems for submanifolds II’, Kyushu J. Math. 54(1) (2000), 103–109.Google Scholar
Topping, P., ‘Relating diameter and mean curvature for submanifolds of Euclidean space’, Comment. Math. Helv. 83 (2008), 539–546.Google Scholar
Wu, J. Y. and Zheng, Y., ‘Relating diameter and mean curvature for Riemannian submanifolds’, Proc. Amer. Math. Soc. 139(11) (2011), 4097–4104.Google Scholar
Yau, S. T., Seminar on Differential Geometry, Problem Section, Annals of Mathematics Studies, 102 (Princeton University Press, Princeton, NJ, 1982), 669–706.Google Scholar