Comptes Rendus
no. 21-22
Differential Geometry
Asymptotic expansion of the Faber–Krahn profile of a compact Riemannian manifold
[Développement asymptotique du profil de Faber–Krahn d'une variété riemannienne compacte]

Présenté par : Étienne Ghys

Olivier Druet1
1 UMPA, ENS Lyon, 46, allée d'Italie, 69364 Lyon cedex 07, France
Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1163-1167.

Nous donnons dans cette Note la preuve d'un résultat bien connu : un développement limité du profil isopérimétrique d'une variété riemannienne donne un développement limité du profil de Faber–Krahn de cette même variété.

The aim of this Note is to give a proof of a well-known fact: an asymptotic expansion of the isoperimetric profile of a Riemannian manifold for small volumes gives an asymptotic expansion of the Faber–Krahn profile for this same Riemannian manifold.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.09.022
Olivier Druet 1

1 UMPA, ENS Lyon, 46, allée d'Italie, 69364 Lyon cedex 07, France 
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     author = {Olivier Druet},
     title = {Asymptotic expansion of the {Faber{\textendash}Krahn} profile of a compact {Riemannian} manifold},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1163--1167},
     publisher = {Elsevier},
     volume = {346},
     number = {21-22},
     year = {2008},
     doi = {10.1016/j.crma.2008.09.022},
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Olivier Druet. Asymptotic expansion of the Faber–Krahn profile of a compact Riemannian manifold. Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1163-1167. doi : 10.1016/j.crma.2008.09.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.09.022/

[1] P. Bérard Spectral Geometry: Direct and Inverse Problems, Lecture Notes in Mathematics, vol. 1207, Springer-Verlag, 1986

[2] I. Chavel Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, vol. 115, Academic Press Inc., 1984

[3] I. Chavel Riemannian Geometry – A Modern Introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, 1993

[4] O. Druet Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc., Volume 130 (2002), pp. 2351-2361

[5] F. Pacard, P. Sicbaldi, Extremal domains for the first eigenvalue of the Laplace–Beltrami operator, Ann. Inst. Fourier, in press

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