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Annals of Global Analysis and Geometry

, Volume 28, Issue 3, pp 257–270 | Cite as

p-Laplace Operator and Diameter of Manifolds

  • Jean-François GrosjeanEmail author
Article

Abstract

Let \({(M^{n},g)}\) be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the p-Laplacian \({\lambda_{1,p}(M)}\) and we prove that the limit of \(p{\sqrt {\lambda_{1,p}(M)}}\) when \(p\rightarrow\infty\) is 2/d(M), where d(M) is the diameter of M. Moreover, if \({(M^{n},g)}\) is an oriented compact hypersurface of the Euclidean space \({\mathbb{R}^{n+1}}\) or \({\mathbb{S}^{n+1}}\), we prove an upper bound of \({\lambda_{1,p}(M)}\) in terms of the largest principal curvature κ over M. As applications of these results, we obtain optimal lower bounds of d(M) in terms of the curvature. In particular, we prove that if M is a hypersurface of \({\mathbb{R}^{n+1}}\) then: \(d(M)\ge\pi/\kappa\).

Keywords

p-Laplacian hypersurfaces curvature geometric inequalities 

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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Institut Élie Cartan (Mathématiques)Université Henri Poincaré Nancy IVandoeuvre-Les-Nancy CedexFrance

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