Existence and uniqueness of constant mean curvature spheres in Sol 3
Résumé
We study the classification of immersed constant mean curvature (CMC) sphe-res in the homogeneous Riemannian 3-manifold Sol 3 , i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H > 1/ √ 3, there exists a unique (up to left translations) immersed CMC H sphere S H in Sol 3 (Hopf-type theorem). Moreover, this sphere S H is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in Sol 3 (Alexandrov-type theorem). The uniqueness parts of these results are also obtained for all real numbers H such that there exists a solution of the isoperimetric problem with mean curvature H.
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