Existence and uniqueness of constant mean curvature spheres in Sol\(_3\) (Q2872160)

It is well known that the two fundamental results in the theory of compact constant mean curvature (CMC) surfaces are the Alexandrov and Hopf theorems. The first one states that the round spheres are the unique embedded compact CMC surfaces in \(\mathbb{R}^3\) whilst the second one states that the round spheres are the unique immersed CMC spheres in \(\mathbb{R}^3\). These two theorems can be generalized to \(3\)-dimensional space forms immediately. Hopf's Theorem was also generalized by \textit{U. Abresch} and \textit{H. Rosenberg} [Acta Math. 193, No. 2, 141--174 (2004; Zbl 1078.53053); Mat. Contemp. 28, 1--28 (2005; Zbl 1118.53036)] to all simply connected homogeneous \(3\)-manifolds with \(4\)-dimensional isometry group: \(H^2\times \mathbb{R}, \; S^2\times \mathbb{R}, \; \text{Nil}_3,\; \widetilde{\mathrm{PSL}_2(\mathbb{R})}\) and Berger spheres. NEWLINENEWLINENEWLINEThe paper under review generalizes Hopf and Alexandrov type theorems to the case of the Sol space, the simply connected homogeneous \(3\)-manifold with \(3\)-dimensional isometry group. Here, the difficulties come from the fact the Sol space has less symmetry and lack of Abresch-Rosenberg quadratic differential, which was crucial in the proofs of other cases.