p-Laplace operator and diameter of manifolds

Let (M-n, g) beacompact 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)root(1,p)(M) when p --> infinity 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 Rn+1 or Sn+1, we prove an upper bound of lambda(1,p)(M) in terms of the largest principal curvature. 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 Rn+1 then: d(M) >= pi/kappa.