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

Abstract

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 Kahler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kahler manifold andP being a compact real hypersurface ofM.