Embedded plateau problem
Author:
Baris Coskunuzer
Journal:
Trans. Amer. Math. Soc. 364 (2012), 12111224
MSC (2010):
Primary 53A10; Secondary 57M50, 49Q05
Published electronically:
October 19, 2011
MathSciNet review:
2869175
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Abstract: We show that if is a simple closed curve bounding an embedded disk in a closed manifold , then there exists a disk in with boundary such that minimizes the area among the embedded disks with boundary . Moreover, is smooth, minimal and embedded everywhere except where the boundary meets the interior of . The same result is also valid for homogeneously regular manifolds with sufficiently convex boundary.
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 [CG]
 D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385446. MR 2188131 (2006g:57030)
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 J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263321. MR 1501590
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 [Fr]
 J. Frehse, Two dimensional variational problems with thin obstacles, Math. Z. 143 (1975), no. 3, 279288. MR 0380550 (52:1450)
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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)
 [GS]
 R. Gulliver and J. Spruck, On embedded minimal surfaces, Ann. of Math. (2) 103 (1976) 331347. MR 0405217 (53:9011)
 [Gu]
 R.D. Gulliver, Regularity of minimizing surfaces of prescribed mean curvature, Ann. of Math. (2) 97 (1973) 275305. MR 0317188 (47:5736)
 [Gui]
 N. Guillen, Optimal regularity for the Signorini problem, Calc. Var. Partial Differential Equations 36 (2009), 533546. MR 2558329 (2010j:35619)
 [HLT]
 J. Hass, J.C. Lagarias and W.P. Thurston, Area inequalities for embedded disks spanning unknotted curves, J. Differential Geom. 68 (2004) no. 1, 129. MR 2152907 (2006e:53015)
 [HS]
 J. Hass and P. Scott, The existence of least area surfaces in manifolds, Trans. Amer. Math. Soc. 310 (1988) no. 1, 87114. MR 965747 (90c:53022)
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 [MY1]
 W. Meeks and S.T. Yau, Topology of threedimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980) 441484. MR 595203 (83d:53045)
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 W. Meeks and S.T. Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982) no. 2, 151168. MR 645492 (83j:53060)
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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/S000299472011054863
PII:
S 00029947(2011)054863
Received by editor(s):
April 28, 2009
Received by editor(s) in revised form:
February 25, 2010
Published electronically:
October 19, 2011
Additional Notes:
The author was partially supported by EUFP7 Grant IRG226062 and TUBITAK Grant 109T685
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
