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

 

The Cauchy problem for improper affine spheres and the Hessian one equation


Authors: Juan A. Aledo, Rosa M. B. Chaves and José A. Gálvez
Journal: Trans. Amer. Math. Soc. 359 (2007), 4183-4208
MSC (2000): Primary 53A15, 35J60; Secondary 53C45
Published electronically: April 6, 2007
MathSciNet review: 2309181
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Abstract: We give a conformal representation for improper affine spheres which is used to solve the Cauchy problem for the Hessian one equation. With this representation, we characterize the geodesics of an improper affine sphere, study its symmetries and classify the helicoidal ones. Finally, we obtain the complete classification of the isolated singularities of the Hessian one Monge-Ampère equation.


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Additional Information

Juan A. Aledo
Affiliation: Departamento de Matemáticas, Universidad de Castilla-La Mancha, 02071 Albacete, Spain
Email: juanangel.aledo@uclm.es

Rosa M. B. Chaves
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, 05315-970 São Paulo-SP, Brazil
Email: rosab@ime.usp.br

José A. Gálvez
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Email: jagalvez@ugr.es

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04378-4
PII: S 0002-9947(07)04378-4
Received by editor(s): June 29, 2005
Published electronically: April 6, 2007
Additional Notes: The first and third authors were partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746. This work was started while the last author was visiting the IME at the University of Sao Paulo. He would like to thank all members of this institution for their hospitality.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.