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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Properly embedded least area planes in Gromov hyperbolic $ 3$-spaces


Author: Baris Coskunuzer
Journal: Proc. Amer. Math. Soc. 136 (2008), 1427-1432
MSC (2000): Primary 53A10; Secondary 57M50
Published electronically: December 7, 2007
MathSciNet review: 2367116
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Abstract: Let $ X$ be a Gromov hyperbolic $ 3$-space with cocompact metric, and $ \Si$ the sphere at infinity of $ X$. We show that for any simple closed curve $ \Gamma$ in $ \Si$, there exists a properly embedded least area plane in $ X$ spanning $ \Gamma$. This gives a positive answer to Gabai's conjecture from 1997. Soma has already proven this conjecture in 2004. Our technique here is simpler and more general, and it can be applied to many similar settings.


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Additional Information

Baris Coskunuzer
Affiliation: Department of Mathematics, Koc University, Istanbul, Turkey
Email: bcoskunuzer@ku.edu.tr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09214-3
PII: S 0002-9939(07)09214-3
Received by editor(s): October 16, 2006
Received by editor(s) in revised form: February 2, 2007
Published electronically: December 7, 2007
Additional Notes: The author was supported by NSF Grant DMS-0603532
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.