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Existence and uniqueness of constant mean curvature spheres in Sol 3

By Benot Daniel and Pablo Mira

Abstract

International audienceWe 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

Topics: Hopf theorem, Alexandrov theorem, Constant mean curvature surfaces, homogeneous 3-manifolds, Sol 3 space, isoperimetric problem, [MATH]Mathematics [math]
Publisher: HAL CCSD
Year: 2013
OAI identifier: oai:HAL:hal-01097503v1
Provided by: HAL - UPEC / UPEM
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