Number of least area planes in Gromov hyperbolic spaces
Author:
Baris Coskunuzer
Journal:
Proc. Amer. Math. Soc. 138 (2010), 29232937
MSC (2010):
Primary 53A10, 57M50
Published electronically:
April 14, 2010
MathSciNet review:
2644904
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Additional Information
Abstract: We show that for a generic simple closed curve in the asymptotic boundary of a Gromov hyperbolic space with cocompact metric , there exists a unique least area plane in such that . This result has interesting topological applications for constructions of canonical dimensional objects in Gromov hyperbolic manifolds.
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 M. Anderson, Complete minimal varieties in hyperbolic space, Invent. Math. 69 (1982) 477494. MR 679768 (84c:53005)
 [A2]
 M. Anderson, Complete minimal hypersurfaces in hyperbolic nmanifolds, Comment. Math. Helv. 58 (1983) 264290. MR 705537 (85e:53076)
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 M. Bestvina, and G. Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991) 469481. MR 1096169 (93j:20076)
 [Ca]
 D. Calegari, Almost continuous extension for taut foliations, Math. Res. Lett. 8 (2001), no. 56, 637640. MR 1879807 (2002k:57070)
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 B. Coskunuzer, Uniform cochains and genuine laminations, Topology 45 (2006) 751784. MR 2236377 (2007c:57028)
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 B. Coskunuzer, Generic Uniqueness of Least Area Planes in Hyperbolic Space, Geom. & Topology 10 (2006) 401412. MR 2224461 (2007a:53005)
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 B. Coskunuzer, Properly Embedded Least Area Planes in Gromov Hyperbolic Spaces, Proc. Amer. Math. Soc. 136 (2008) 14271432. MR 2367116 (2008j:53010)
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 B. Coskunuzer, On the Number of Solutions to Asymptotic Plateau Problem, eprint; math.DG/0505593
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 B. Coskunuzer, Least Area Planes in Hyperbolic Space are Properly Embedded, Indiana Univ. Math. J. 58 (2009) 381392. MR 2504417 (2010b:53012)
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 B. Coskunuzer, Asymptotic Plateau problem, eprint; arXiv:0907.0552
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 S.R. Fenley, Foliations with good geometry, J. Amer. Math. Soc. 12 (1999), no. 3, 619676. MR 1674739 (2000b:57041)
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 D. Gabai, On the geometric and topological rigidity of hyperbolic manifolds, J. Amer. Math. Soc. 10 (1997) 3774. MR 1354958 (97h:57028)
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 U. Lang, Asymptotic Plateau problem in Gromov hyperbolic manifolds, Calc. Var. Partial Differential Equations 16 (2003) 3146. MR 1951491 (2003m:49064)
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Additional Information
Baris Coskunuzer
Affiliation:
Department of Mathematics, Koc University, Sariyer, Istanbul 34450, Turkey
Email:
bcoskunuzer@ku.edu.tr
DOI:
http://dx.doi.org/10.1090/S0002993910103086
Received by editor(s):
October 15, 2009
Received by editor(s) in revised form:
December 2, 2009
Published electronically:
April 14, 2010
Additional Notes:
The author is partially supported by EUFP7 Grant IRG226062, TUBITAK Grant 107T642 and a TUBAGEBIP Award
Communicated by:
Alexander N. Dranishnikov
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
