Sharp growth estimates for warping functions in multiply warped product manifolds

TL;DR

This paper establishes sharp growth estimates for warping functions in multiply warped product manifolds immersed in Riemannian manifolds, revealing a dichotomy between constancy and infinity of these functions based on PDE methods.

Contribution

It generalizes previous work by providing sharp inequalities linking mean curvature and sectional curvatures through analytic growth conditions of warping functions.

Findings

Dichotomy between constancy and infinity of warping functions.

Sharp inequalities relating mean curvature and sectional curvatures.

Applications demonstrating growth estimates in geometric analysis.

Abstract

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\times_{f_2} N_2 \times \cdots \times _{f_k} N_k\, $ into a Riemannian manifold. Generalizing the earlier work of the authors in [{Glasg. Math. J. 51 (2009) 579-592], 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.