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
Fulltext PDF Free Access
Abstract 
References 
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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.
 [A1]
Michael
T. Anderson, Complete minimal varieties in hyperbolic space,
Invent. Math. 69 (1982), no. 3, 477–494. MR 679768
(84c:53005), 10.1007/BF01389365
 [A2]
Michael
T. Anderson, Complete minimal hypersurfaces in hyperbolic
𝑛manifolds, Comment. Math. Helv. 58 (1983),
no. 2, 264–290. MR 705537
(85e:53076), 10.1007/BF02564636
 [BM]
Mladen
Bestvina and Geoffrey
Mess, The boundary of negatively curved
groups, J. Amer. Math. Soc.
4 (1991), no. 3,
469–481. MR 1096169
(93j:20076), 10.1090/S08940347199110961691
 [Ca]
Danny
Calegari, Almost continuous extension for taut foliations,
Math. Res. Lett. 8 (2001), no. 56, 637–640. MR 1879807
(2002k:57070), 10.4310/MRL.2001.v8.n5.a5
 [Co1]
Baris
Coskunuzer, Uniform 1cochains and genuine laminations,
Topology 45 (2006), no. 4, 751–784. MR 2236377
(2007c:57028), 10.1016/j.top.2006.03.002
 [Co2]
Baris
Coskunuzer, Generic uniqueness of least area planes in hyperbolic
space, Geom. Topol. 10 (2006), 401–412
(electronic). MR
2224461 (2007a:53005), 10.2140/gt.2006.10.401
 [Co3]
Baris
Coskunuzer, Properly embedded least area planes in
Gromov hyperbolic 3spaces, Proc. Amer. Math.
Soc. 136 (2008), no. 4, 1427–1432. MR 2367116
(2008j:53010), 10.1090/S0002993907092143
 [Co4]
B. Coskunuzer, On the Number of Solutions to Asymptotic Plateau Problem, eprint; math.DG/0505593
 [Co5]
Baris
Coskunuzer, Least area planes in hyperbolic 3space are properly
embedded, Indiana Univ. Math. J. 58 (2009),
no. 1, 381–392. MR 2504417
(2010b:53012), 10.1512/iumj.2009.58.3447
 [Co6]
B. Coskunuzer, Asymptotic Plateau problem, eprint; arXiv:0907.0552
 [Fe]
Sérgio
R. Fenley, Foliations with good
geometry, J. Amer. Math. Soc.
12 (1999), no. 3,
619–676. MR 1674739
(2000b:57041), 10.1090/S0894034799003045
 [Ga]
David
Gabai, On the geometric and topological
rigidity of hyperbolic 3manifolds, J. Amer.
Math. Soc. 10 (1997), no. 1, 37–74. MR 1354958
(97h:57028), 10.1090/S0894034797002063
 [GK]
David
Gabai and William
H. Kazez, Group negative curvature for 3manifolds with genuine
laminations, Geom. Topol. 2 (1998), 65–77
(electronic). MR
1619168 (99e:57023), 10.2140/gt.1998.2.65
 [Gr]
M.
Gromov, Hyperbolic groups, Essays in group theory, Math. Sci.
Res. Inst. Publ., vol. 8, Springer, New York, 1987,
pp. 75–263. MR 919829
(89e:20070), 10.1007/9781461395867_3
 [HL]
Robert
Hardt and FangHua
Lin, Regularity at infinity for areaminimizing hypersurfaces in
hyperbolic space, Invent. Math. 88 (1987),
no. 1, 217–224. MR 877013
(88m:49033), 10.1007/BF01405098
 [HS]
Joel
Hass and Peter
Scott, The existence of least area surfaces
in 3manifolds, Trans. Amer. Math. Soc.
310 (1988), no. 1,
87–114. MR
965747 (90c:53022), 10.1090/S00029947198809657476
 [L]
Urs
Lang, The asymptotic Plateau problem in Gromov hyperbolic
manifolds, Calc. Var. Partial Differential Equations
16 (2003), no. 1, 31–46. MR 1951491
(2003m:49064), 10.1007/s005260100140
 [MY]
William
H. Meeks III and Shing
Tung Yau, The classical Plateau problem and the topology of
threedimensional manifolds. The embedding of the solution given by
DouglasMorrey and an analytic proof of Dehn’s lemma, Topology
21 (1982), no. 4, 409–442. MR 670745
(84g:53016), 10.1016/00409383(82)900210
 [So1]
Teruhiko
Soma, Existence of least area planes in hyperbolic 3space with
cocompact metric, Topology 43 (2004), no. 3,
705–716. MR 2041639
(2005a:57017), 10.1016/j.top.2003.10.006
 [So2]
Teruhiko
Soma, Least area planes in Gromov hyperbolic 3spaces with
cocompact metric, Geom. Dedicata 112 (2005),
123–128. MR 2163893
(2006d:53007), 10.1007/s107110043241x
 [A1]
 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)
 [BM]
 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)
 [Co1]
 B. Coskunuzer, Uniform cochains and genuine laminations, Topology 45 (2006) 751784. MR 2236377 (2007c:57028)
 [Co2]
 B. Coskunuzer, Generic Uniqueness of Least Area Planes in Hyperbolic Space, Geom. & Topology 10 (2006) 401412. MR 2224461 (2007a:53005)
 [Co3]
 B. Coskunuzer, Properly Embedded Least Area Planes in Gromov Hyperbolic Spaces, Proc. Amer. Math. Soc. 136 (2008) 14271432. MR 2367116 (2008j:53010)
 [Co4]
 B. Coskunuzer, On the Number of Solutions to Asymptotic Plateau Problem, eprint; math.DG/0505593
 [Co5]
 B. Coskunuzer, Least Area Planes in Hyperbolic Space are Properly Embedded, Indiana Univ. Math. J. 58 (2009) 381392. MR 2504417 (2010b:53012)
 [Co6]
 B. Coskunuzer, Asymptotic Plateau problem, eprint; arXiv:0907.0552
 [Fe]
 S.R. Fenley, Foliations with good geometry, J. Amer. Math. Soc. 12 (1999), no. 3, 619676. MR 1674739 (2000b:57041)
 [Ga]
 D. Gabai, On the geometric and topological rigidity of hyperbolic manifolds, J. Amer. Math. Soc. 10 (1997) 3774. MR 1354958 (97h:57028)
 [GK]
 D. Gabai, W.H. Kazez, Group negative curvature for manifolds with genuine laminations, Geom. Topol. 2 (1998) 6577. MR 1619168 (99e:57023)
 [Gr]
 M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ. 8 Springer (1987) 75263. MR 919829 (89e:20070)
 [HL]
 R. Hardt and F.H. Lin, Regularity at infinity for absolutely area minimizing hypersurfaces in hyperbolic space, Invent. Math. 88 (1987) 217224. MR 877013 (88m:49033)
 [HS]
 J. Hass and P. Scott, The Existence of Least Area Surfaces in manifolds, Trans. AMS 310 (1988) 87114. MR 965747 (90c:53022)
 [L]
 U. Lang, Asymptotic Plateau problem in Gromov hyperbolic manifolds, Calc. Var. Partial Differential Equations 16 (2003) 3146. MR 1951491 (2003m:49064)
 [MY]
 W. Meeks and S.T. Yau, The classical Plateau problem and the topology of three manifolds, Topology 21 (1982) 409442. MR 670745 (84g:53016)
 [So1]
 T. Soma, Existence of least area planes in hyperbolic space with cocompact metric, Topology 43 (2004) 705716. MR 2041639 (2005a:57017)
 [So2]
 T. Soma, Least area planes in Gromov hyperbolic spaces with cocompact metric, Geom. Dedicata 112 (2005) 123128. MR 2163893 (2006d:53007)
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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.
