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Surfaces with parallel mean curvature in $ \mathbb{C}P^n\times\mathbb{R}$ and $ \mathbb{C}H^n\times\mathbb{R}$


Authors: Dorel Fetcu and Harold Rosenberg
Journal: Trans. Amer. Math. Soc. 366 (2014), 75-94
MSC (2010): Primary 53A10, 53C42
DOI: https://doi.org/10.1090/S0002-9947-2013-05704-2
Published electronically: May 31, 2013
MathSciNet review: 3118391
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Abstract: We consider surfaces with parallel mean curvature vector (pmc surfaces) in $ \mathbb{C}P^n\times \mathbb{R}$ and $ \mathbb{C}H^n\times \mathbb{R}$, and, more generally, in cosymplectic space forms. We introduce a holomorphic quadratic differential on such surfaces. This is then used in order to show that the anti-invariant pmc $ 2$-spheres of a $ 5$-dimensional non-flat cosymplectic space form of product type are actually the embedded rotational spheres $ S_H^2\subset \bar M^2\times \mathbb{R}$ of Hsiang and Pedrosa, where $ \bar M^2$ is a complete simply-connected surface with constant curvature. When the ambient space is a cosymplectic space form of product type and its dimension is greater than $ 5$, we prove that an immersed non-minimal non-pseudo-umbilical anti-invariant $ 2$-sphere lies in a product space $ \bar M^4\times \mathbb{R}$, where $ \bar M^4$ is a space form. We also provide a reduction of codimension theorem for the pmc surfaces of a non-flat cosymplectic space form.


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Additional Information

Dorel Fetcu
Affiliation: Department of Mathematics and Informatics, “Gh. Asachi” Technical University of Iasi, Bd. Carol I no. 11, 700506 Iasi, Romania
Email: dfetcu@math.tuiasi.ro

Harold Rosenberg
Affiliation: Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, Brasil
Email: rosen@impa.br

DOI: https://doi.org/10.1090/S0002-9947-2013-05704-2
Keywords: Surfaces with parallel mean curvature vector, cosymplectic space forms, quadratic forms
Received by editor(s): November 22, 2010
Received by editor(s) in revised form: August 30, 2011
Published electronically: May 31, 2013
Additional Notes: The first author was supported by a Post-Doctoral Fellowship “Pós-Doutorado Sênior (PDS)” offered by FAPERJ, Brazil.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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