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
DOI:
https://doi.org/10.1090/S0002-9947-2011-05346-8
Published electronically:
October 3, 2011
MathSciNet review:
2846353
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Abstract | References | Similar Articles | Additional Information
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
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