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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

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Mathematical Citation Quotient (MCQ) 2018: 1.55

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Volume 2013, Issue 685


Existence and uniqueness of constant mean curvature spheres in

Benoît Daniel
  • Université Paris-Est, Laboratoire d'Analyse et de Mathématiques Appliquées, CNRS, UMR 8050, 61 avenue du Général de Gaulle, 94010 Créteil, France
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/ Pablo Mira
  • Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30203 Cartagena, Murcia, Spain
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Published Online: 2012-03-23 | DOI: https://doi.org/10.1515/crelle-2012-0016


We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold , i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every , there exists a unique (up to left translations) immersed CMC H sphere SH in (Hopf-type theorem). Moreover, this sphere SH is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in (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.

About the article

Received: 2009-09-18

Revised: 2012-01-10

Published Online: 2012-03-23

Published in Print: 2013-12-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2013, Issue 685, Pages 1–32, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2012-0016.

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