Biharmonic submanifolds with parallel mean curvature in ${\mathbb{S}}^n\times \mathbb{R}$

We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces $M^n(c)\times \mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in ${\mathbb{S}}^n(c)\times \mathbb{R}$.