Eigenvalue Estimates for Submanifolds with Locally Bounded Mean Curvature

Abstract

We present a method to obtain lower bounds for firstDirichlet eigenvalue in terms of vector fields with positivedivergence. Applying this to the gradient of a distance functionwe obtain estimates of eigenvalue of balls inside the cut locus and of domains Ω ⊂ M ∩ BN(p, r) in submanifolds M ⊂ϕNwith locally bounded mean curvature. Forsubmanifolds of Hadamard manifolds with bounded mean curvaturethese lower bounds depend only on the dimension of the submanifold and the bound on its mean curvature.