Comparison of Laplace Beltrami Operator Eigenvalues on Riemannian Manifolds

Abstract

Let $\Delta_{g}$ be the Laplace Beltrami operator on a manifold $M$ with Dirichlet (resp.,Neumann) boundary conditions. We compare the spectrum of on a Riemannian manifold for Neumann boundary condition and Dirichlet boundary condition . Then we construct aneffective method of obtaining small eigenvalues for Neumann's problem.