Bounds for the first Dirichlet eigenvalue attained at an infinite family of Riemannian manifolds

LetM be a compact Riemannian manifold with smooth boundary M. We get bounds for the first eigenvalue of the Dirichlet eigenvalue problem onM in terms of bounds of the sectional curvature ofM and the normal curvatures of M. We discuss the equality, which is attained precisely on certain model spaces defined by J. H. Eschenburg. We also get analog results for Kähler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kähler manifold andP being a compact real hypersurface ofM.Work partially supported by a DGICYT Grant No. PB94-0972 and by the E.C. Contract CHRX-CT92-0050 GADGET II.