Classification of rotational special Weingarten surfaces of minimal type in {mathbb{S}^2 times mathbb{R}} and {mathbb{H}^2 times mathbb{R}}

In this paper we classify the complete rotational special Weingarten surfaces in ${mathbb{S}^2 times mathbb{R}}$ and ${mathbb{H}^2 times mathbb{R}}$ ; i.e. rotational surfaces in ${mathbb{S}^2 times mathbb{R}}$ and ${mathbb{H}^2 times mathbb{R}}$ whose mean curvature H and extrinsic curvature K _e satisfy H = f(H ² ? K _e ), for some function ${f in mathcal{C}^1([0,+infty))}$ such that f(0) = 0 and 4x(f′(x))² < 1 for any x ≥ 0. Furthermore we show the existence of non-complete examples of such surfaces.