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Integral curvature estimates for F-stable hypersurfaces

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Abstract.

We consider immersed hypersurfaces in euclidean \(% \mathbb{R}^{{n + 1}} % $% which are stable with respect to an elliptic parametric functional with integrand F = F(N) depending on normal directions only. We prove an integral curvature estimate provided that F is sufficiently close to the area integrand, extending the classical estimate of Schoen, Simon and Yau [19] for stable minimal hypersurfaces in \(% \mathbb{R}^{{n + 1}} % $%, as well as the pointwise estimate of Simon [22] for F-minimizing hypersurfaces. As a crucial point of our analysis we derive a generalized Simons inequality for the laplacian of the length of a weighted second fundamental form with respect to an abstract metric associated with F. As an application, we obtain a new Bernstein result for complete F-stable hypersurfaces of dimension \(n \leq 5\).

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  1. Institut für Mathematik, Universität Duisburg-Essen, 47048, Duisburg, Germany

    Sven Winklmann

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  1. Sven Winklmann
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Correspondence to Sven Winklmann.

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Received: 14 July 2003, Accepted: 13 September 2004, Published online: 10 December 2004

Mathematics Subject Classification:

53C42, 49Q10, 35J60

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Winklmann, S. Integral curvature estimates for F-stable hypersurfaces. Calc. Var. 23, 391–414 (2005). https://doi.org/10.1007/s00526-004-0306-5

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  • DOI: https://doi.org/10.1007/s00526-004-0306-5

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