Stabilité de l'inégalité de Faber-Krahn en courbure de Ricci positive  [ Stability of the Faber-Krahn inequality in positive Ricci curvature ]
Annales de l'Institut Fourier, Volume 55 (2005) no. 2, p. 353-372

P. Bérard and D. Meyer proved a Faber-Krahn inequality for domains in compact manifolds with positive Ricci curvature. We prove stability results for this inequality

P. Bérard et D. Meyer ont démontré une inégalité du type Faber-Krahn pour les domaines d'une variété compacte à courbure de Ricci positive. Nous démontrons des résultats de stabilité associés à cette inégalité.

DOI : https://doi.org/10.5802/aif.2101
Classification:  53C20,  53C24,  58C40,  51K
Keywords: Riemannian Geometry, Gromov-Hausdorff distance, Faber-Krahn inequality, convex domains 
@article{AIF_2005__55_2_353_0,
     author = {Bertrand, J\'er\^ome},
     title = {Stabilit\'e de l'in\'egalit\'e de Faber-Krahn en courbure de Ricci positive},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {2},
     year = {2005},
     pages = {353-372},
     doi = {10.5802/aif.2101},
     zbl = {1080.53032},
     mrnumber = {2147894},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2005__55_2_353_0}
}

Bertrand, Jérôme. Stabilité de l'inégalité de Faber-Krahn en courbure de Ricci positive. Annales de l'Institut Fourier, Volume 55 (2005) no. 2, pp. 353-372. doi : 10.5802/aif.2101. http://www.numdam.org/item/AIF_2005__55_2_353_0/

[1] M. T. Anderson Metrics of positive Ricci curvature with large diameter, Manuscripta Math., Tome 68 (1990) no. 4, pp. 405-415 | MR 1068264 | Zbl 0711.53036

[2] A.I. Ávila Stability results for the first eigenvalue of the Laplacian on domains in space forms, J. Math. Anal. Appl., Tome 267 (2002) no. 2, pp. 760-774 | Article | MR 1888036 | Zbl 1189.35200 | Zbl 01749683

[3] P. Bérard; D. Meyer Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup., Tome 15 (1982) no. 3, pp. 513-541 | Numdam | MR 690651 | Zbl 0527.35020

[4] J. Bertrand Pincement spectral en courbure de Ricci positive ((à paraître)) | Zbl 1118.53018

[5] J. Bertrand Pincement spectral en courbure positive (2003) (Thèse de doctorat)

[6] I. Chavel; E. A. Feldman Spectra of domains in compact manifolds, J. Funct. Anal., Tome 30 (1975) no. 2, pp. 198-222 | MR 515225 | Zbl 0392.58016

[7] I. Chavel Eigenvalues in Riemannian geometry, Academic Press Inc., Orlando, FL, Pure and Applied Mathematics, Tome 115 (1984) | MR 768584 | Zbl 0551.53001

[8] J. Cheeger; T. H. Colding Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math., Tome 144 (1996) no. 1, pp. 189-237 | Article | MR 1405949 | Zbl 0865.53037

[9] Shiu Yuen Cheng Eigenvalue comparison theorems and its geometric applications, Math. Z., Tome 143 (1975) no. 3, pp. 289-297 | Article | MR 378001 | Zbl 0329.53035

[10] S. Gallot Inégalités isopérimétriques, courbure de Ricci et invariants géométriques, C. R. Acad. Sci. Paris Sér. I Math., Tome 296 (1983) no. 8, pp. 365-368 | MR 699164 | Zbl 0535.53035

[11] M. Gromov Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser Boston Inc., Boston MA, Progress in Mathematics, Tome 152 | MR 1699320 | Zbl 0953.53002

[12] K. Grove; P. V. Petersen A pinching theorem for homotopy spheres, J. Amer. Math. Soc., Tome 3 (1990) no. 3, pp. 671-677 | Article | MR 1049696 | Zbl 0717.53025

[13] K. Grove; K. Shiohama A generalized sphere theorem, Ann. Math., Tome 106 (1977) no. 2, pp. 201-211 | MR 500705 | Zbl 0341.53029

[14] P. Li; Shing Tung Yau Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator, Amer. Math. Soc., Providence R.I (Proc. Sympos. Pure Math.) Tome XXXVI (1979), pp. 205-239 | Zbl 0441.58014

[15] J. Rauch; M. Taylor Potential and scattering theory on wildly perturbed domains, J. Funct. Anal., Tome 18 (1975), pp. 27-59 | Article | MR 377303 | Zbl 0293.35056

[16] R. C. Reilly Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., Tome 26 (1977) no. 3, pp. 459-472 | Article | MR 474149 | Zbl 0391.53019