Isoperimetric control of the spectrum of a compact hypersurface

Bruno Colbois 1 , Ahmad El Soufi 2 ,  and Alexandre Girouard 3
  • 1 Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, Case postale 158, 2009 Neuchâtel, Switzerland
  • 2 Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 6083, Université Frano̧is Rabelais de Tours, Parc de Grandmont, 37200 Tours, France
  • 3 Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, Case postale 158, 2009 Neuchâtel, Switzerland

Abstract.

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.

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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