Rotational symmetry of Weingarten spheres in homogeneous three-manifolds

Let $M$ be a simply connected homogeneous three-manifold with isometry group of dimension $4$, and let $\mathrm{\Sigma }$ be any compact surface of genus zero immersed in $M$ whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation $\mathrm{\Phi }(H,K_e,K)=0$. In this paper we prove that $\mathrm{\Sigma }$ is a sphere of revolution, provided that the unique inextendible rotational surface $S$ in $M$ that satisfies this equation and touches its rotation axis orthogonally has bounded second fundamental form. In particular, we prove that: (i) any elliptic Weingarten sphere immersed in ${\mathbb{H}}^2\times \mathbb{R}$ is a rotational sphere. (ii) Any sphere of constant positive extrinsic curvature immersed in $M$ is a rotational sphere, and (iii) Any immersed sphere in $M$ that satisfies an elliptic Weingarten equation $H=\varphi (H^2-K_e)\ge a>0$ with $\varphi$ bounded, is a rotational sphere. As a very particular case of this last result, we recover the Abresch-Rosenberg classification of constant mean curvature spheres in $M$.