Existence and uniqueness of constant mean curvature spheres in

✍ Scribed by Daniel, Benoît; Mira, Pablo

• Book ID: 121873018
• Publisher: Walter de Gruyter GmbH & Co. KG
• Year: 2013
• Tongue: English
• Weight: 414 KB
• Volume: 2013
• Category: Article
• ISSN: 0075-4102
• DOI: 10.1515/crelle-2012-0016

✦ Synopsis

We study the classification of immersed constant mean curvature (CMC) spheres 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= p 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 .