IMMERSION OF MANIFOLDS WITH UNBOUNDED IMAGE AND A MODIFIED MAXIMUM PRINCIPLE OF YAU

Let N be a complete Riemannian manifold isometrically immersed into a Hadamard manifold M. We show that the immersion cannot be bounded if the mean curvature of the immersed manifold is small compared with the curvature of M and the Laplacian of the distance function on N grows at most linearly. The latter condition is satisfied if the Ricci curvature of N does not approach -infinity too fast. The main tool in the proof is a modification of Yau's maximum principle.