In this article, we consider compact surfaces $\Sigma$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $\Gamma=\partial\Sigma\subset \mathbb{ M}_0= \mathbb{ M} \times_f\{ 0\}$ is transversal to the slice $\mathbb{ M}_0$ of the warped product $ \mathbb{ M}\times_f\mathbb{ R} $, here $ \mathbb{ M} $ denotes a Hadamard surface. We obtain height estimate for a such surface $\Sigma$ having positive constant mean curvature involving the area of a part of $\Sigma$ above of $ \mathbb{ M} _0$ and the volume it bounds. Also we give general conditions for the existence of rotationally-invariant topological spheres having positive constant mean curvature $H$ in the warped product $\mathbb{ H}\times_f\mathbb{ R}$, where $\mathbb{ H}$ denotes the hyperbolic disc. Finally we present a non-trivial example of such spheres.
Categories: math.DG