Existence and uniqueness of constant mean curvature spheres in Sol_3

Abstract

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/(\sqrt{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.

36 pages, 2 figures; minor changes

References in corpus (4)

• The Isoperimetric Profile of a Noncompact Riemannian Manifold for Small Volumes
• The Gauss map of minimal surfaces in the Heisenberg group
• Totally umbilic surfaces in homogeneous 3-manifolds
• On stable constant mean curvature surfaces in S2 X R and H2 X R

Cited by in corpus (2)

• Hypersurfaces with constant anisotropic mean curvatures
• The Gauss map of surfaces in ~PSL_2(R)