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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space


Authors: Isabel Fernández and Pablo Mira
Journal: Trans. Amer. Math. Soc. 361 (2009), 5737-5752
MSC (2000): Primary 53A10
Published electronically: June 22, 2009
MathSciNet review: 2529912
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Abstract: We classify the entire minimal vertical graphs in the Heisenberg group $ {\rm Nil_3}$ endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in $ {\rm Nil}_3$, is given in terms of the Abresch-Rosenberg holomorphic differential for minimal surfaces in $ {\rm Nil}_3$.


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Additional Information

Isabel Fernández
Affiliation: Departamento de Matematica Aplicada I, Universidad de Sevilla, E-41012 Sevilla, Spain
Email: isafer@us.es

Pablo Mira
Affiliation: Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, E-30203 Cartagena, Murcia, Spain
Email: pablo.mira@upct.es

DOI: https://doi.org/10.1090/S0002-9947-09-04645-5
Keywords: Minimal graphs, Bernstein problem, holomorphic quadratic differential, Heisenberg group
Received by editor(s): May 15, 2007
Published electronically: June 22, 2009
Additional Notes: The first author was partially supported by MEC-FEDER Grant No. MTM2007-64504 and Regional J. Andalucia Grants P06-FQM-01642 and FQM 325
The second author was partially supported by MEC-FEDER, Grant No. MTM2007-65249 and the Programme in Support of Excellence Groups of Murcia, by Fund. Seneca, reference 04540/GERM/OG
Article copyright: © Copyright 2009 American Mathematical Society