Abstract
In this paper, our aim is to establish optimal upper estimates for the first positive eigenvalue of a Jacobi type operator, which is a suitable extension of the linearized operators of the higher order mean curvatures of a closed hypersurface immersed either in spherical or in hyperbolical spaces.
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References
Alencar H., Colares A.G.: Integral formulas for the r-mean curvature linearized operator of a hypersurface. Ann. Glob. Anal. Geom. 16, 203–220 (1998)
Alencar H., do Carmo M., Marques F.: Upper bounds for the first eigenvalue of the linearized operator L r and some applications. Ill. J. Math. 45, 851–863 (2001)
Alencar H., do Carmo M., Rosenberg H.: On the first eigenvalue of the linearized operator of the r-th mean curvature of a hypersurface. Ann. Glob. Anal. Geom. 11, 387–395 (1993)
Alías L.J., Malacarne J.M.: On the first eingenvalue of the linearized operator of the highter order mean curvature for closed hypersurfaces in spaces forms. Ill. J. Math. 48, 219–240 (2004)
Barbosa J.L.M., Colares A.G.: Stability of hypersurfaces with constant r-mean curvature. Ann. Glob. Anal. Geom. 15, 277–297 (1997)
Cheng S.Y., Yau S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 (1977)
El Soufi A., Ilias S.: Une inégalité du type “Reilly” pour les sous-variétés de l’espace hyperbolique. Comment. Math. Helv. 67, 167–181 (1992)
de Lima H.F., Velásquez M.A.L., Sousa A.F.: On the stability of hypersurfaces in space forms. J. Math. Anal. Appl. 406, 147–157 (2013)
Grosjean J.-F.: Extrinsic upper bounds for the first eigenvalue of elliptc operators. Hokkaido Math. J. 33(2), 319–339 (2004)
Heintze E.: Extrinsic upper bound for \({\lambda_{1}}\). Math. Ann. 280, 389–402 (1988)
Reilly R.C.: On the first eigenvalue of the Laplacian operator for compact submanifolds of Euclidean space. Comment. Math. Helv. 52, 525–533 (1977)
Rosenberg H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117, 211–239 (1993)
Veeravalli A.R.: On the first Laplacian eigenvalue and the center of gravity of compact hypersurfaces. Comment. Math. Helv. 76, 155–160 (2001)
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de Lima, H.F., de Sousa, A.F., dos Santos, F.R. et al. Optimal Upper Estimates for the First Eigenvalue of a Jacobi Type Operator in Spherical and Hyperbolical Spaces. Mediterr. J. Math. 13, 3907–3919 (2016). https://doi.org/10.1007/s00009-016-0723-7
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DOI: https://doi.org/10.1007/s00009-016-0723-7