Metric upper bounds for Steklov and Laplace eigenvalues
[Bornes supérieures métriques pour les valeurs propres de Steklov et de Laplace]
Colbois, Bruno1 ; Girouard, Alexandre2
1 Université de Neuchâtel Institut de Mathématiques Rue Emile-Argand 11 CH-2000 Neuchâtel, Switzerland
2 Département de mathématiques et de statistique Pavillon Alexandre-Vachon, Université Laval Québec, QC, G1V 0A6, Canada
Annales de l'Institut Fourier, Online first, 18 p.

We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as packing and volume growth constants of the boundary and its distortion. Its proof is based on metric-measure space techniques. The second bound is in terms of the extrinsic diameter of the boundary and its injectivity radius. It is obtained from a concentration inequality, akin to Gromov–Milman concentration for closed manifolds. By applying these bounds to cylinders over closed manifolds, we obtain bounds for eigenvalues of the Laplace operator, in the spirit of Berger–Croke. For a family of manifolds that has uniformly bounded volume and boundary of fixed intrinsic geometry, we deduce that a large first nonzero Steklov eigenvalue implies that each boundary component is contained in a ball of small extrinsic radius.

Nous obtenons deux bornes supérieures pour les valeurs propres de Steklov d’une variété riemannienne compacte à bord. La première fait intervenir le volume de la variété et de son bord, des constantes d’empilement et de croissance du bord ainsi que sa distorsion. La preuve utilise une technique provenant de la théorie des espaces métriques mesurés. La deuxième borne dépend du diamètre extrinsèque du bord et de son rayon d’injectivité et découle d’une inégalité de concentration à la Gromov–Milman. En appliquant ces bornes à des cylindres, on obtient des bornes pour les valeurs propres du laplacien sur des variétés fermées, semblables à celles de Berger–Croke. Pour des variétés dont le volume est uniformément borné et dont le bord est de géométrie intrinsèque prescrite, nous déduisons qu’une grande valeur propre de Steklov implique que chaque composante du bord est contenue dans une boule extrinsèque de petit rayon.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3742
Classification : 58C40
Keywords: Steklov spectrum, geometric eigenvalue bounds
Mots-clés : spectre de Steklov, borne géométrique pour les valeurs propres

Colbois, Bruno 1 ; Girouard, Alexandre 2

1 Université de Neuchâtel Institut de Mathématiques Rue Emile-Argand 11 CH-2000 Neuchâtel, Switzerland
2 Département de mathématiques et de statistique Pavillon Alexandre-Vachon, Université Laval Québec, QC, G1V 0A6, Canada 
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Colbois, Bruno; Girouard, Alexandre. Metric upper bounds for Steklov and Laplace eigenvalues. Annales de l'Institut Fourier, Online first, 18 p.

[1] Berger, M. Une inégalité universelle pour la première valeur propre du laplacien, Bull. Soc. Math. Fr., Volume 107 (1979) no. 1, pp. 3-9 | Numdam | Zbl | DOI | MR

[2] Brisson, J. Tubular excision and Steklov eigenvalues, J. Geom. Anal., Volume 32 (2022) no. 5, 166, 24 pages | Zbl | DOI | MR

[3] Brisson, J. Multiple tubular excisions and large Steklov eigenvalues, Ann. Global Anal. Geom., Volume 65 (2024) no. 2, 18, 19 pages | Zbl | DOI | MR

[4] Cianci, D.; Girouard, Alexandre Large spectral gaps for Steklov eigenvalues under volume constraints and under localized conformal deformations, Ann. Global Anal. Geom., Volume 54 (2018) no. 4, pp. 529-539 | Zbl | DOI | MR

[5] Colbois, B.; Dodziuk, J. Riemannian metrics with large λ 1 , Proc. Am. Math. Soc., Volume 122 (1994) no. 3, pp. 905-906 | Zbl | DOI | MR

[6] Colbois, B.; El Soufi, A.; Girouard, Alexandre Isoperimetric control of the Steklov spectrum, J. Funct. Anal., Volume 261 (2011) no. 5, pp. 1384-1399 | Zbl | DOI | MR

[7] Colbois, B.; El Soufi, A.; Girouard, Alexandre Isoperimetric control of the spectrum of a compact hypersurface, J. Reine Angew. Math., Volume 683 (2013), pp. 49-65 | Zbl | MR

[8] Colbois, B.; El Soufi, A.; Girouard, Alexandre Compact manifolds with fixed boundary and large Steklov eigenvalues, Proc. Am. Math. Soc., Volume 147 (2019) no. 9, pp. 3813-3827 | Zbl | DOI | MR

[9] Colbois, B.; Girouard, Alexandre The spectral gap of graphs and Steklov eigenvalues on surfaces, Electron. Res. Announc. Math. Sci., Volume 21 (2014), pp. 19-27 | Zbl | DOI | MR

[10] Colbois, B.; Girouard, Alexandre; Gittins, K. Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space, J. Geom. Anal., Volume 29 (2019) no. 2, pp. 1811-1834 | Zbl | DOI | MR

[11] Colbois, B.; Girouard, Alexandre; Gordon, C.; Sher, D. Some recent developments on the Steklov eigenvalue problem, Rev. Mat. Complut., Volume 37 (2024) no. 1, pp. 1-161 | Zbl | DOI | MR

[12] Colbois, B.; Girouard, Alexandre; Hassannezhad, A. The Steklov and Laplacian spectra of Riemannian manifolds with boundary, J. Funct. Anal., Volume 278 (2020) no. 6, 108409, 38 pages | Zbl | DOI | MR

[13] Colbois, B.; Gittins, K. Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index, Differ. Geom. Appl., Volume 78 (2021), 101777, 21 pages | Zbl | DOI | MR

[14] Colbois, B.; Maerten, D. Eigenvalues estimate for the Neumann problem of a bounded domain, J. Geom. Anal., Volume 18 (2008) no. 4, pp. 1022-1032 | Zbl | DOI | MR

[15] Colbois, B.; Savo, A. Large eigenvalues and concentration, Pac. J. Math., Volume 249 (2011) no. 2, pp. 271-290 | Zbl | DOI | MR

[16] Colbois, B.; Verma, S. Sharp Steklov upper bound for submanifolds of revolution, J. Geom. Anal., Volume 31 (2021) no. 11, pp. 11214-11225 | Zbl | DOI | MR

[17] Croke, C. B. Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Éc. Norm. Supér. (4), Volume 13 (1980) no. 4, pp. 419-435 | DOI | Numdam | Zbl | MR

[18] Fraser, A.; Schoen, R. Some results on higher eigenvalue optimization, Calc. Var. Partial Differ. Equ., Volume 59 (2020) no. 5, 151, 22 pages | Zbl | DOI | MR

[19] Girouard, Alexandre; Karpukhin, M.; Lagacé, J. Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems, Geom. Funct. Anal., Volume 31 (2021) no. 3, pp. 513-561 | Zbl | DOI | MR

[20] Girouard, Alexandre; Lagacé, J. Large Steklov eigenvalues via homogenisation on manifolds, Invent. Math., Volume 226 (2021) no. 3, pp. 1011-1056 | MR | Zbl | DOI

[21] Girouard, Alexandre; Polterovich, I. Spectral geometry of the Steklov problem (survey article), J. Spectr. Theory, Volume 7 (2017) no. 2, pp. 321-359 | MR | Zbl | DOI

[22] Girouard, Alexandre; Polymerakis, Panagiotis Large Steklov eigenvalues under volume constraints, J. Geom. Anal., Volume 34 (2024) no. 10, 316, 19 pages | MR | Zbl | DOI

[23] Grigoryan, A.; Netrusov, Y.; Yau, S.-T. Eigenvalues of elliptic operators and geometric applications, Surveys in differential geometry. Vol. IX (Surveys in Differential Geometry), International Press, 2004 no. 9, pp. 147-217 | Zbl | DOI | MR

[24] Gromov, M.; Milman, V. D. and A topological application of the isoperimetric inequality, Am. J. Math., Volume 105 (1983) no. 4, pp. 843-854 | Zbl | DOI | MR

[25] Ilias, S.; Makhoul, O. A Reilly inequality for the first Steklov eigenvalue, Differ. Geom. Appl., Volume 29 (2011) no. 5, pp. 699-708 | Zbl | DOI | MR

[26] Kokarev, G. Berger’s inequality in the presence of upper sectional curvature bound, Int. Math. Res. Not., Volume 2022 (2022) no. 15, pp. 11262-11303 | Zbl | DOI | MR

[27] Korevaar, N. Upper bounds for eigenvalues of conformal metrics, J. Differ. Geom., Volume 37 (1993) no. 1, pp. 73-93 | DOI | Zbl | MR

[28] Xiong, C. Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds, J. Funct. Anal., Volume 275 (2018) no. 12, pp. 3245-3258 | Zbl | DOI | MR

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