Isoperimetric control of the spectrum of a compact hypersurface

Bruno Colbois (1) , Ahmad El Soufi (2) , Alexandre Girouard (1)

1 Laboratoire de Mathématiques
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

Upper bounds for the eigenvalues of the Laplace-Beltrami operator on a hypersurface bounding a domain in some ambient Riemannian manifold are given in terms of the isoperimetric ratio of the domain. These results are applied to the extrinsic geometry of isometric embeddings.

Mots clés: Laplacian; eigenvalue; submanifold; isoperimetric ratio

Domaines: Géométrie métrique [math.MG]

Bruno Colbois, Ahmad El Soufi, Alexandre Girouard. Isoperimetric control of the spectrum of a compact hypersurface. Journal für die reine und angewandte Mathematik, 2013, 683, pp.49--65. ⟨10.1515/crelle-2012-0008⟩. ⟨hal-00497630v2⟩