Niveau: Supérieur, Doctorat, Bac+8
Existence and uniqueness of constant mean curvature spheres in Sol3 Benoıt Daniela and Pablo Mirab a Universite Paris 12, Departement de Mathematiques, UFR des Sciences et Technolo- gies, 61 avenue du General de Gaulle, 94010 Creteil cedex, France e-mail: b Departamento de Matematica Aplicada y Estadıstica, Universidad Politecnica de Cartagena, E-30203 Cartagena, Murcia, Spain. e-mail: AMS Subject Classification: 53A10, 53C42 Keywords: Constant mean curvature surfaces, homogeneous 3-manifolds, Sol3 space, Hopf theorem, Alexandrov theorem, isoperimetric problem. Abstract We study the classification of immersed constant mean curvature (CMC) sphe- res in the homogeneous Riemannian 3-manifold Sol3, 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 SH in Sol3 (Hopf-type theorem). Moreover, this sphere SH is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in Sol3 (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.
- alexandrov theorem
- called hopf differential
- sol3
- mean curvature
- lie group
- there exists