Total mean curvature surfaces in the product space ${\mathbb{S}}^n\times \mathbb{R}$ and applications

The total mean curvature functional for submanifolds into the Riemannian product space ${\mathbb{S}}^n\times \mathbb{R}$ is considered and its first variational formula is presented. Later on, two second order differential operators are defined and a nice integral inequality relating both of them is proved. Finally we prove our main result: an integral inequality for closed stationary $\mathcal{H}$-surfaces in ${\mathbb{S}}^n\times \mathbb{R}$, characterizing the cases where the equality is attained.