Abstract
We prove Reilly-type upper bounds for different types of eigenvalue problems on submanifolds of Euclidean spaces with density. This includes the eigenvalues of Paneitz-like operators as well as three types of generalized Steklov problems. In the case without density, the equality cases are discussed and we prove some stability results for hypersurfaces which derive from a general pinching result about the moment of inertia.
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Alencar, H., Carmo, M. P., Rosenberg, H.: On the first eigenvalue of Linearized operator of the r-th mean curvature of a hypersurface. Ann. Glob. Anal. Geom. 11, 387–395 (1993)
Aubry, E., Grosjean, J.F.: Spectrum of hypersurfaces with small extrinsic radius or large λ1 in Euclidean spaces. J. Funct. Anal. 271(5), 1213–1242 (2016)
Aubry, E., Grosjean, J.F., Roth, J.: Hypersurfaces with small extrinsic radius or large λ1, in Euclidean spaces. arXiv:1009.2010
Alias, L.J., Malacarne, J.M.: On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms. Illinois J. Math. 48(1), 219–240 (2004)
Auchmuty, G.: Steklov Eigenproblems and the representation of solutions of elliptic boundary value problems. Numer. Funct. Anal. Optim. 25(3–4), 321–348 (2004)
Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math, pp 177–206. Springer, Berlin (1985)
Barbosa, J.L.M., Colares, A.G.: Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15, 277–297 (1997)
Batista, M., Santos, J.I.: The first Stekloff eigenvalue in weighted Riemannian manifolds. arXiv:1504.02630 (2015)
Batista, M., Cavalcante, M.P., Pyo, J.: Some isomperimetric inequalities and eigenvalue estimates in weighted manifolds. J. Math. Anal. Appl. 419(1), 617–626 (2014)
Branson, T.: Differential operators canonically associated to a conformal structure. Math. Scand. 57, 293–345 (1985)
Brendle, S.: Embedded self-similar shrinkers of genus 0. Ann. Math. 183, 715–728 (2016)
Buoso, D., Provenzano, L.: On the Eigenvalues of a Biharmonic Steklov Problem: Integral Methods in Science and Engineering: Theoretical and Computational Advances. Birkhauser, Cambridge (2015)
Caubet, F., Kateb D., Le Louër, F.: Shape sensitivity analysis for elastic structures with generalized impedance boundary conditions of the Wentzell type—application to compliance minimization, Hal:01525249
Chen, D., Li, H.: The sharp estimates for the first eigenvalue of Paneitz operator on 4-dimensional submanifolds. arXiv:1010.3102
Colbois, B., Grosjean, J.F.: A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of Euclidean spaces. Comment. Math. Helv. 82, 175–195 (2007)
Dambrine, M., Kateb, D., Lamboley, J.: An extremal eigenvalue problem for the Wentzell-Laplace operator. Ann. I. H. Poincaré 33(2), 409–450 (2016)
Djadli, Z., Hebey, E., Ledoux, M.: Paneitz-type operators and applications. Duke Math. J. 104(1), 129–169 (2000)
Domingo-Juan, M.C., Miquel, V.: Reilly’s type inequality for the Laplacian associated to a density related with shrinkers for MCF. arXiv:1503.01332
El Soufi, A., Ilias, S.: Une inégalité de type “Reilly” pour les sous-variétés de l’espace hyperbolique. Comment. Math. Helv. 67(2), 167–181 (1992)
El Soufi, A., Harrell, E.M. II, Ilias, S.: Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds. Trans. Amer. Math. Soc. 361 (5), 2337–2350 (2009)
Grosjean, J. F.: Upper bounds for the first eigenvalue of the Laplacian on compact manifolds. Pac. J. Math. 206(1), 93–111 (2002)
Guan, P. F., Shen, S.: A rigidity theorem for hypersurfaces in higher dimensional space forms. In: Analysis, Complex Geometry, and Mathematical Physics: in Honor of Duong H. Phong, 61–65, Contemp. Math., vol. 644. American Mathematical Society, Providence (2015)
Hsiung, C. C.: Some integral formulae for closed hypersurfaces. Math. Scand. 2, 286–294 (1954)
Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)
Hu, Y., Xu, H., Zhao, E.: First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures. Ann. Glob. Anal. Geom. 1–13 (2015)
Ilias, S., Makhoul, O.: A Reilly inequality for the first Steklov eigenvalue. Differ. Geom. Appl. 29(5), 699–708 (2011)
Lichnerowicz, A.: Variétés riemanniennes à tenseurr C non négatif. C.R. Acad. Sc. Paris Ser. A 271, A650–A653 (1970)
Paneitz, S.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds preprint (1983)
Paneitz, S.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. SIGMA 4, article 036 (2008)
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.: Extrinsic radius pinching for hypersurfaces of space forms. Differ. Geom. Appl. 25(5), 485–499 (2007)
Roth, J.: Extrinsic radius pinching in space forms with nonnegative sectional curvature. Math. Z. 258(1), 227–240 (2008)
Roth, J.: Upper bounds for the first eigenvalue of the Laplacian in terms of anisiotropic mean curvatures. Results Math. 64(3–4), 383–403 (2013)
Roth, J.: General Reilly-type inequalities for submanifolds of weighted Euclidean spaces. Colloq. Math. 144(1), 127–136 (2016)
Roth, J., Scheuer, J.: Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space. Ann. Glob. Anal. Geom. 51(3), 287–304 (2017)
Steklov, W.: Sur les problèmes fondamentaux de la physique mathémnatique (suite et fin). Ann. Sci. École Norm. Sup. (3) 319, 455–490 (1902)
Wang, Q., Xia, C.: Sharp bounds for the first non-zero Stekloff eigenvalues. J. Funct. Anal. 257, 2635–2644 (2009)
Wei, G., Wylie, W.: Comparison geometry for the Bakry-Émery Ricci curvature. J. Differ. Geom. 83, 377–405 (2009)
Xia, C., Wang, Q.: Eigenvalues of the Wentzell-Laplace operator and of the fourth order Steklov problems. Journal Differential Equations, 264(10), 6486–6506. arXiv:1506.03780
Yang, P., Xu, X.: Positivity of Paneitz operators. Discrete Cont. Dyn. Syst. 7(2), 329–342 (2001)
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Roth, J. Reilly-Type Inequalities for Paneitz and Steklov Eigenvalues. Potential Anal 53, 773–798 (2020). https://doi.org/10.1007/s11118-019-09787-7
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DOI: https://doi.org/10.1007/s11118-019-09787-7