A first eigenvalue estimate for embedded hypersurfaces

Suppose that M is a compact orientable hypersurface embedded in a compact n-dimensional orientable Riemannian manifold N. Suppose that the Ricci curvature of N is bounded below by a positive constant k. We show that 2 λ 1 > k − ( n − 1 ) max M | H | where λ 1 is the first eigenvalue of the Laplacian of M and H is the mean curvature of M.