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Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space

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Abstract

We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to r-stability as well as to almost-Einstein hypersurfaces.

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References

  1. Alencar, H., DoCarmo, M., Colares, A.: Stable hypersurfaces with constant scalar curvature. Math. Z. 213(1), 117–131 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alencar, H., Do Carmo, M., Marques, F.C.: Upper bounds for the first eigenvalue of the operator \(L_r\) and some applications. Ill. J. Math. 45(3), 851–863 (2001)

  3. 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(4), 387–395 (1993)

  4. Alias, L., Malacarne, M.: On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms. Ill. J. Math. 48(1), 219–240 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Aubry, E.: Finiteness of \(\pi _{1}\) and geometric inequalities in almost positive Ricci curvature. Ann. Sci. Éc. Norm. Supér. (4) 40(4), 675–695 (2007)

  6. Aubry, E.: Diameter pinching in almost positive Ricci curvature. Comment. Math. Helv. 84(2), 223–233 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Aubry, E., Grosjean, J.F.: Spectrum of hypersurfaces with small extrinsic radius or large \(\lambda _1\) in Euclidean spaces. J. Funct. Anal. 271(5), 1213–1242 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Aubry, E., Grosjean, J.F., Roth, J.: Hypersurfaces with small extrinsic radius or large \(\lambda _1\) in Euclidean spaces. Preprint arXiv:1009.2010

  9. Barbosa, J., Colares, A.: Stability of hypersurfaces with constant r-mean curvature. Ann. Glob. Anal. Geom. 15(3), 277–297 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Barbosa, J., Do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Z. 185(3), 339–353 (1984)

  11. Barbosa, J., Do Carmo, M., Eschenburg, J.H.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z. 197(1), 123–138 (1988)

  12. Bleecker, D., Weiner, J.: Extrinsic bound on \(\lambda _1\) of \(\delta \) on a compact manifold. Comment. Math. Helv. 51, 601–609 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheng, X.: A generalization of almost-Schur lemma for closed Riemannian manifolds. Ann. Glob. Anal. Geom. 43, 153–160 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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(1), 175–195 (2007)

  15. De Lellis, C., Topping, P.: Almost-Schur lemma. Calc. Var. Partial Differ. Equ. 43(3–4), 347–354 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ge, Y., Wang, G.: An almost Schur theorem on 4-dimensional manifolds. Proc. Am. Math. Soc. 140(3), 1041–1044 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gerhardt, C.: Curvature problems. In: Series in Geometry and Topology, vol. 39. International Press of Boston Inc., Somerville (2006)

  19. Grosjean, J.F.: Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds. Pac. J. Math. 206(1), 93–112 (2002)

  20. Grosjean, J.F.: Extrinsic upper bounds for the first eigenvalue of elliptic operators. Hokkaido Math. J. 33(2), 319–339 (2004)

  21. Grosjean, J.F., Roth, J.: Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces. Math. Z. 271(1–2), 469–488 (2012)

  22. Hardy, G., Littlewood, J., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1952)

  23. Heintze, E.: Extrinsic upper bounds for \(\lambda _1\). Math. Ann. 280(3), 389–402 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hiroshima, T.: Construction of the Green function on Riemannian manifold using harmonic coordinates. J. Math. Tokyo Univ. 36(1), 1–30 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hoffmann, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27(6), 715–727 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hu, Y., Xu, H., Zhao, E.: First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures. Ann. Glob. Anal. Geom. 48(1), 23–35 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Michael, J., Simon, L.: Sobolev and mean-value inequalities on generalized submanifolds of \(\mathbb{R}^{n}\). Commun. Pure Appl. Math. 26(3), 361–379 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  28. Reilly, R.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8(3), 465–477 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  29. Reilly, R.: On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space. Comment. Math. Helv. 52(4), 525–533 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoam. 3(3–4), 447–453 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Roth, J.: Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of the Euclidean space. Ann. Glob. Anal. Geom. 33(3), 293–306 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Roth, J.: A remark on almost umbilical hypersurfaces. Arch. Math. 49(1), 1–7 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Roth, J.: Upper bounds for the first eigenvalue of the Laplacian of hypersurfaces in terms of anisotropic mean curvatures. Result. Math. 64(3–4), 383–403 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Roth, J.: General Reilly-type inequalities for submanifolds of weighted Euclidean spaces. Colloq. Math. 144(1), 127–136 (2016)

    MathSciNet  MATH  Google Scholar 

  35. Scheuer, J.: Quantitative oscillation estimates for almost-umbilical closed hypersurfaces in Euclidean space. Bull. Aust. Math. Soc. 92(1), 133–144 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Takahashi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Jpn. 18(4), 380–385 (1966)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Julian Scheuer.

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Roth, J., Scheuer, J. Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space. Ann Glob Anal Geom 51, 287–304 (2017). https://doi.org/10.1007/s10455-016-9535-z

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