Surfaces with parallel mean curvature in $\mathbb {C}P^n\times \mathbb {R}$ and $\mathbb {C}H^n\times \mathbb {R}$
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- by Dorel Fetcu and Harold Rosenberg PDF
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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.References
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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
- MR Author ID: 150570
- Email: rosen@impa.br
- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3118391