Extrinsic estimates for eigenvalues of the Laplace operator

Abstract

For a bounded domain in a complete Riemannian manifold $M^n$ isometrically immersed in a Euclidean space, we derive extrinsic estimates for eigenvalues of the Dirichlet eigenvalue problem of the Laplace operator, which depend on the mean curvature of the immersion. Further, we also obtain an upper bound for the ${(k+1)}^{\text{th}}$ eigenvalue, which is best possible in the meaning of order on $k$.