GROWTH ESTIMATES FOR WARPING FUNCTIONS AND THEIR GEOMETRIC APPLICATIONS

摘要

By applying Wei, Li and Wu’s notion (given in ‘Generalizations ofnthe uniformization theorem and Bochner’s method in p-harmonic geometry’, Comm.nMath. Anal. Conf., vol. 01, 2008, pp. 46–68) and method (given in ‘Sharp estimatesnon A-harmonic functions with applications in biharmonic maps, preprint) and bynmodifying the proof of a general inequality given by Chen in ‘On isometric minimalnimmersion from warped products into space forms’ (Proc. Edinb. Math. Soc., vol. 45,n2002, pp. 579–587), we establish some simple relations between geometric estimatesn(the mean curvature of an isometric immersion of a warped product and sectionalncurvatures of an ambient m-manifold ˜M mnc bounded from above by a non-positivennumber c) and analytic estimates (the growth of the warping function). We find andichotomy between constancy and ‘infinity’ of the warping functions on completennon-compact Riemannian manifolds for an appropriate isometric immersion. Severalnapplications of our growth estimates are also presented. In particular, we prove thatnif f is an Lq function on a complete non-compact Riemannian manifold N1 for somenq > 1, then for any Riemannian manifold N2 the warped product N1 ×f N2 does notnadmit a minimal immersion into any non-positively curved Riemannian manifold.Wenalso show that both the geometric curvature estimates and the analytic function growthnestimates in this paper are sharp.