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 $\mathbb{S}^n(c)\times\mathbb{R}$. 
Dorel Fetcu, Cezar Oniciuc, and Harold Rosenberg
math.DG