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P LAPLACE OPERATOR AND DIAMETER OF MANIFOLDS

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Niveau: Supérieur, Doctorat, Bac+8
P -LAPLACE OPERATOR AND DIAMETER OF MANIFOLDS Jean-Franc¸ois GROSJEAN November 4, 2005 GROSJEAN Jean-Franc¸ois, Institut Elie Cartan (Mathematiques), Universite Henri Poincare Nancy I, B.P. 239, F-54506 VANDOEUVRE-LES-NANCY CEDEX, FRANCE. E-mail: Abstract Let (Mn, g) be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the p-Laplacian ?1,p(M) and we prove that the limit of p √ ?1,p(M) when p ? ∞ is 2/d(M) where d(M) is the diameter of M . Moreover if (Mn, g) is an oriented compact hypersurface of the Euclidean space Rn+1 or Sn+1, we prove an upper bound of ?1,p(M) in term of the largest principal curvature ? over M . As applications of these results we obtain optimal lower bounds of d(M) in term of the curvature. In particular we prove that if M is a hypersurface of Rn+1 then : d(M) ≥ pi/?. Mathematics Subject Classification (2000): 53A07, 53C21. 1

  • compact manifold

  • eigenfunction associated

  • without boundary

  • associated respec- tively

  • max x??

  • boundary

  • ∞?∞ ≤

  • ∂?

  • compact riemannian

  • lim infpk?∞


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Language English
P -LAPLACE OPERATOR AND DIAMETER OF MANIFOLDS
Jean-Franc¸oisGROSJEAN November 4, 2005
´ GROSJEANJean-Fran¸cois,InstitutElieCartan(Mathe´matiques),Universit´eHenri Poincar´eNancyI,B.P.239,F-54506VANDOEUVRE-LES-NANCYCEDEX,FRANCE. E-mail: grosjean@iecn.u-nancy.fr
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 λ 1 ,p ( M ) and we prove that the limit of p p λ 1 ,p ( M ) when p → ∞ 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 R n +1 or S n +1 , we prove an upper bound of λ 1 p ( M ) in term of the largest , principal curvature κ over M . As applications of these results we obtain optimal lower bounds of d ( M ) in term of the curvature. In particular we prove that if M is a hypersurface of R n +1 then : d ( M ) π/κ .
Mathematics Subject Classification (2000): 53A07, 53C21.
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