GROWTH ESTIMATES FOR WARPING FUNCTIONS AND THEIR GEOMETRIC APPLICATIONS

Abstract

AbstractBy 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 -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 $\tilde M^m_c$ 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 Lq function on a complete non-compact Riemannian manifold N1 for some q > 1, then for any Riemannian manifold N2 the warped product N1 ×fN2 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.