Eigenvalue Estimates for Submanifolds with Locally Bounded Mean Curvature

摘要

We present a method to obtain lower bounds for first Dirichlet eigenvalue in terms of vector fields with positive divergence. Applying this to the gradient of a distance function we obtain estimates of eigenvalue of balls inside the cut locus and of domains Ω is contained in M ∩ B_N(p, r) in submanifolds M (is contained in)_φ N with locally bounded mean curvature. For submanifolds of Hadamard manifolds with bounded mean curvature these lower bounds depend only on the dimension of the submanifold and the bound on its mean curvature.