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Comptes Rendus Mathématique
Volume 357, n° 2
pages 180-187 (février 2019)
Doi : 10.1016/j.crma.2019.01.006
Received : 13 April 2018 ;  accepted : 21 January 2019
Eigenvalue and gap estimates of isometric immersions for the Dirichlet-to-Neumann operator acting on p -forms
Estimations de valeurs propres et du gap de l'opérateur de Dirichlet-à-Neumann agissant sur les p -formes pour les immersions isométriques
 

Deborah Michel
 Laboratoire de mathématiques Raphaël-Salem, UMR 6085 CNRS–Université de Rouen, avenue de l'Université, BP 12, Technopôle du Madrillet, 76801 Saint-Étienne-du-Rouvray, France 

Abstract

In this paper, we study the first eigenvalue of the Dirichlet-to-Neumann operator acting on differential forms of a Riemannian manifold with boundary isometrically immersed in some Euclidean space. We give a lower bound of the integral energy of p -forms in terms of its first eigenvalue associated with  -forms. We also find a lower bound for the gap between two consecutive first eigenvalues in terms of the curvature of the boundary.

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Résumé

Dans cet article, nous étudions la première valeur propre de l'opérateur de Dirichlet-à-Neumann agissant sur les formes différentielles d'une variété riemannienne à bord plongée isométriquement dans un espace euclidien. Nous obtenons une borne inférieure de l'énergie des p -formes en termes de sa première valeur propre associée aux  -formes. Nous trouvons aussi une borne inférieure pour l'écart entre deux premières valeurs propres consécutives par rapport à la courbure de la frontière.

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