GROWTH ESTIMATES FOR WARPING FUNCTIONS AND THEIR GEOMETRIC APPLICATIONS

By applying Wei, Li and Wu's notion (given in 'Generalizations of the uniformization theorem and Bochner's method in p-harmonic geometry', Comm. Math. Anal. Conf, vol. 01, 2008, pp. 46-68) and method (given in 'Sharp estimates on A-harmonic functions with applications in biharmonic maps, preprint) and by modifying the proof of a general inequality given by Chen in 'On isometric minimal immersion from warped products into space forms' (Proc. Edinb. Math. Soc., vol. 45, 2002, pp. 579-587), we establish some simple relations between geometric estimates (the mean curvature of an isometric immersion of a warped product and sectional curvatures of an ambient m-manifold (M) over tilde (m)(c) bounded from above by a non-positive number c) and analytic estimates (the growth of the warping function). We find a dichotomy between constancy and 'infinity' of the warping functions on complete non-compact Riemannian manifolds for an appropriate isometric immersion. Several applications of our growth estimates are also presented. In particular, we prove that if f is an L(q) function on a complete non-compact Riemannian manifold N(1) for some q > 1, then for any Riemannian manifold N(2) the warped product N(1) x (f) N(2) does not admit a minimal immersion into any non-positively curved Riemannian manifold. We also show that both the geometric curvature estimates and the analytic function growth estimates in this paper are sharp.