Submanifolds immersed in a warped product: Rigidity and nonexistence

Abstract:

In this paper, we deal with $n$-dimensional submanifolds immersed in a warped product space of the type $I\times _fM^{n+p}$ whose warping function $f$ has convex logarithm. Assuming that such a submanifold $\psi :\Sigma ^n\rightarrow I\times _{f}M^{n+p}$ is either closed, stochastically complete, or complete with nonnegative Ricci curvature, and that its support function $\langle \vec {H},\partial _{t}\rangle$ is constant (where $\vec {H}$ stands for the mean curvature vector field of $\psi$ and $\partial _t$ denotes the unit vector field tangent to the interval $I\subset \mathbb R$), we prove that $\psi (\Sigma )$ must be contained in a slice of $I\times _fM^{n+p}$. As a consequence of our rigidity results, when $p=1$ we obtain nonexistence results concerning minimal submanifolds immersed in such an ambient space.