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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Surfaces with parallel mean curvature vector in $ \mathbb{S}^2\times\mathbb{S}^2$ and $ \mathbb{H}^2\times\mathbb{H}^2$


Authors: Francisco Torralbo and Francisco Urbano
Journal: Trans. Amer. Math. Soc. 364 (2012), 785-813
MSC (2010): Primary 53A10; Secondary 53B35
Published electronically: October 3, 2011
MathSciNet review: 2846353
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Abstract: Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in $ \mathbb{S}^2\times \mathbb{S}^2$ and $ \mathbb{H}^2\times \mathbb{H}^2$ are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of $ \mathbb{S}^2\times \mathbb{R}$ and $ \mathbb{H}^2\times \mathbb{R}$ is established. Using this, surfaces with vanishing Hopf differentials (in particular, spheres with parallel mean curvature vector) are classified and a rigidity result for constant mean curvature surfaces of $ \mathbb{S}^2\times \mathbb{R}$ and $ \mathbb{H}^2\times \mathbb{R}$ is proved.


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Additional Information

Francisco Torralbo
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Email: ftorralbo@ugr.es

Francisco Urbano
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Email: furbano@ugr.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05346-8
PII: S 0002-9947(2011)05346-8
Received by editor(s): December 30, 2008
Received by editor(s) in revised form: October 27, 2009, and February 15, 2010
Published electronically: October 3, 2011
Additional Notes: This research was partially supported by an MCyT-Feder research project MTM2007-61775 and a Junta Andalucĭa Grant P06-FQM-01642.
Article copyright: © Copyright 2011 American Mathematical Society