Rotational symmetry of Weingarten spheres in homogeneous three-manifolds

TL;DR

This paper proves that certain elliptic Weingarten spheres in homogeneous three-manifolds are rotational, extending classification results and showing all such spheres are rotational under specified conditions.

Contribution

It establishes that elliptic Weingarten spheres in homogeneous three-manifolds are rotational, generalizing known classifications and providing new conditions for rotational symmetry.

Findings

Elliptic Weingarten spheres in $ ext{H}^2 imes ext{R}$ are rotational.

Spheres with constant positive extrinsic curvature are rotational.

Spheres satisfying specific elliptic Weingarten equations are rotational.

Abstract

Let $M$ be a simply connected homogeneous three-manifold with isometry group of dimension $4$, and let $\Sigma$ be any compact surface of genus zero immersed in $M$ whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation $\Phi(H,K_e,K)=0$. In this paper we prove that $\Sigma$ 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 $H=\phi(H^2-K_e)\geq a>0$ with $\phi$ 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 $M$.