Eigenvalue estimates for submanifolds with locally bounded mean   curvature
  We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean curvature of the submanifold. For submanifolds fo Hadamard manifolds these lower bounds depend only on the dimension and mean curvature of the submanifold. 
G. Pacelli Bessa and J. Fabio Montenegro