Rotational symmetry of Weingarten spheres in homogeneous three-manifolds

Abstract

Abstract Let M be a simply connected homogeneous three-manifold with isometry group of dimension 4, and let Σ be any compact surface of genus zero immersed in M whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation Φ ⁢ ( H , K e , K ) = 0 {\Phi(H,K_{e},K)=0} . In this paper we prove that Σ 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 ℍ 2 × ℝ {\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. (iii) Any immersed sphere in M that satisfies an elliptic Weingarten equation H = ϕ ⁢ ( H 2 - K e ) ≥ a > 0 {H=\phi(H^{2}-K_{e})\geq a>0} with ϕ 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.