SHARP GROWTH ESTIMATES FOR WARPING FUNCTIONS IN MULTIPLY WARPED PRODUCT MANIFOLDS

By applying an average method in PDE, we obtain a dichotomy between "constancy" and "infinity" of the warping functions on complete noncompact Riemannian manifolds for an appropriate isometric immersion of a multiply warped product manifold N-1 x (f2) N-2 x ... x (fk) N-k into a Riemannian manifold. Generalizing the earlier work of the authors in [9], we establish sharp inequalities between the mean curvature of the immersion and the sectional curvatures of the ambient manifold under the influence of quantities of a purely analytic nature (the growth of the warping functions). Several applications of our growth estimates are also presented.