Existence and uniqueness of constant mean curvature spheres in Sol 3

Benot Daniel (1) , Pablo Mira (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 UPCT - Universidad Politécnica de Cartagena / Technical University of Cartagena

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.

Mots clés: Constant mean curvature surfaces; homogeneous 3-manifolds; Sol 3 space; Hopf theorem; Alexandrov theorem; isoperimetric problem

Domaines: Mathématiques [math]

Benot Daniel, Pablo Mira. Existence and uniqueness of constant mean curvature spheres in Sol 3. Journal für die reine und angewandte Mathematik, 2013, pp.1-32. ⟨hal-01097503⟩