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 .

中文翻译：

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Weingarten球在均匀三流形中的旋转对称性

令M为等距三流形的简单连接，其等距群为4维，等式Σ为零的任何紧实表面，浸入M的均值，外在曲率和高斯曲率均满足光滑椭圆关系Φ⁢（H，K e， K）= 0 {\ Phi（H，K_ {e}，K）= 0}。在本文中，我们证明Σ是一个旋转球，只要M中满足该方程式并且正交地接触其旋转轴的唯一不可扩展的旋转表面S具有第二基本形式的边界。特别地，我们证明：（i）浸入ℍ2×ℝ{\ mathbb {H} ^ {2} \ times \ mathbb {R}}中的任何椭圆形Weingarten球都是旋转球。（ii）浸入M的任何具有恒定正外在曲率的球都是旋转球。（iii）M中满足椭圆Weingarten方程H = ϕ（H 2-K e）≥a>的任何浸入球体 0 {H = \ phi（H ^ {2} -K_ {e}）\ geq a> 0}以ϕ为界，是一个旋转球体。作为最后一个结果的特例，我们恢复了M中的恒定平均曲率球的Abresch-Rosenberg分类。