Abstract
We prove upper bounds for the first eigenvalue of the Laplacian of hypersurfaces of Euclidean space involving anisotropic mean curvatures. Then, we study the equality case and its stability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alexandrov A.D.: A characteristic property of spheres. Ann. Math. Pura Appl. 58, 303–315 (1962)
Colbois B., Grosjean J.F.: A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space. Comment. Math. Helv. 82, 175–195 (2007)
Grosjean J.F., Roth J.: Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces. Math. Z. 271(1), 469–488 (2012)
He Y., Li H.: Integral formula of Minkowski type and new characterization of the Wulff Shape. Acta Math. Sin. 24(3), 697–704 (2008)
He Y., Li H.: A new variational characterization of the Wulff shape. Differ. Geom. Appl. 26(4), 377–390 (2008)
He Y., Li H., Ma H., Ge J.: Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures. Indiana Univ. Math. J. 58(2), 853–868 (2009)
Hoffman D., Spruck D.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27, 715–727 (1974)
Hoffman D., Spruck J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Erratum Comm. Pure and Appl. Math. 28, 765–766 (1975)
Hsiung C.C.: Some integral formulae for closed hypersurfaces. Math. Scand. 2, 286–294 (1954)
Koiso M., Palmer B.: Geometry and stability of surfaces with constant anisotropic mean curvature. Indiana Univ. Math. J. 54, 1817–1852 (2005)
Michael J.H., Simon L.M.: Sobolev and mean-value inequalities on generalized submanifolds of \({\mathbb{R}^n}\) . Commun. Pure Appl. Math. 26, 361–379 (1973)
Montiel S., Ros A.: Compact hypersurfaces: the Alexandrov theorem for higher order mean curvature. Pitman Monogr. Surv. Pure Appl. Math. 52, 279–296 (1991)
Onat L.: Some characterizations of the Wulff shape. C. R. Math. Acad. Sci. Paris 348(17–18), 997–1000 (2010)
Palmer B.: Stability of the Wulff shape. Proc. Am. Math. Soc. 126(2), 3661–3667 (1998)
Reilly R.C.: On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space. Comment. Math. Helv. 52, 525–533 (1977)
Roth J.: Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of Euclidean space. Ann. Glob. Anal. Geom. 33(3), 293–306 (2008)
Sakai, T.: Riemannian geometry. A.M.S. Transl. Math. Monogr. 149 (1996)
Takahashi T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18(4), 380–385 (1966)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Roth, J. Upper Bounds for the First Eigenvalue of the Laplacian of Hypersurfaces in terms of Anisotropic Mean Curvatures. Results. Math. 64, 383–403 (2013). https://doi.org/10.1007/s00025-013-0322-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-013-0322-x