Illinois Journal of Mathematics

On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms

Luis J. Alías and J. Miguel Malacarne

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Abstract

n this paper we derive sharp upper bounds for the first positive eigenvalue of the linearized operator of the higher order mean curvature of a closed hypersurface immersed into a Riemannian space form. Our bounds are extrinsic in the sense that they are given in terms of the higher order mean curvatures and the center(s) of gravity of the hypersurface, and they extend previous bounds recently given by Veeravalli [Ve] for the first positive eigenvalue of the Laplacian operator.

Article information

Source
Illinois J. Math., Volume 48, Number 1 (2004), 219-240.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258136182

Digital Object Identifier
doi:10.1215/ijm/1258136182

Mathematical Reviews number (MathSciNet)
MR2048223

Zentralblatt MATH identifier
1038.53061

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds

Citation

Alías, Luis J.; Malacarne, J. Miguel. On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms. Illinois J. Math. 48 (2004), no. 1, 219--240. doi:10.1215/ijm/1258136182. https://projecteuclid.org/euclid.ijm/1258136182


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