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.
 [Al]
Hans
Wilhelm Alt, Verzweigungspunkte von 𝐻Flächen.
II, Math. Ann. 201 (1973), 33–55 (German). MR 0331195
(48 #9529)
 [AS]
Frederick
J. Almgren Jr. and Leon
Simon, Existence of embedded solutions of Plateau’s
problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6
(1979), no. 3, 447–495. MR 553794
(81d:49025)
 [At]
Ioannis
Athanasopoulos, Coincidence set of minimal surfaces for the thin
obstacle, Manuscripta Math. 42 (1983), no. 23,
199–209. MR
701203 (85a:49048), http://dx.doi.org/10.1007/BF01169583
 [CG]
Danny
Calegari and David
Gabai, Shrinkwrapping and the taming of
hyperbolic 3manifolds, J. Amer. Math. Soc.
19 (2006), no. 2,
385–446. MR 2188131
(2006g:57030), http://dx.doi.org/10.1090/S0894034705005138
 [CM]
Tobias
H. Colding and William
P. Minicozzi II, Minimal surfaces, Courant Lecture Notes in
Mathematics, vol. 4, New York University, Courant Institute of
Mathematical Sciences, New York, 1999. MR 1683966
(2002b:49072)
 [Do]
Jesse
Douglas, Solution of the problem of
Plateau, Trans. Amer. Math. Soc.
33 (1931), no. 1,
263–321. MR
1501590, http://dx.doi.org/10.1090/S00029947193115015909
 [EWW]
Tobias
Ekholm, Brian
White, and Daniel
Wienholtz, Embeddedness of minimal surfaces with total boundary
curvature at most 4𝜋, Ann. of Math. (2) 155
(2002), no. 1, 209–234. MR 1888799
(2003f:53010), http://dx.doi.org/10.2307/3062155
 [Fe]
Herbert
Federer, Geometric measure theory, Die Grundlehren der
mathematischen Wissenschaften, Band 153, SpringerVerlag New York Inc., New
York, 1969. MR
0257325 (41 #1976)
 [Fr]
Jens
Frehse, Two dimensional variational problems with thin
obstacles, Math. Z. 143 (1975), no. 3,
279–288. MR 0380550
(52 #1450)
 [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), http://dx.doi.org/10.1090/S0894034797002063
 [GS]
Robert
Gulliver and Joel
Spruck, On embedded minimal surfaces, Ann. of Math. (2)
103 (1976), no. 2, 331–347. MR 0405217
(53 #9011)
 [Gu]
Robert
D. Gulliver II, Regularity of minimizing surfaces of prescribed
mean curvature, Ann. of Math. (2) 97 (1973),
275–305. MR 0317188
(47 #5736)
 [Gui]
Nestor
Guillen, Optimal regularity for the Signorini problem, Calc.
Var. Partial Differential Equations 36 (2009), no. 4,
533–546. MR 2558329
(2010j:35619), http://dx.doi.org/10.1007/s0052600902425
 [HLT]
Joel
Hass, Jeffrey
C. Lagarias, and William
P. Thurston, Area inequalities for embedded disks spanning
unknotted curves, J. Differential Geom. 68 (2004),
no. 1, 1–29. MR 2152907
(2006e:53015)
 [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), http://dx.doi.org/10.1090/S00029947198809657476
 [Le]
Hans
Lewy, On the coincidence set in variational inequalities, J.
Differential Geometry 6 (1972), 497–501. Collection
of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth
birthdays. MR
0320343 (47 #8882)
 [Mo]
Charles
B. Morrey Jr., The problem of Plateau on a Riemannian
manifold, Ann. of Math. (2) 49 (1948), 807–851.
MR
0027137 (10,259f)
 [MY1]
William
H. Meeks III and Shing
Tung Yau, Topology of threedimensional manifolds and the embedding
problems in minimal surface theory, Ann. of Math. (2)
112 (1980), no. 3, 441–484. MR 595203
(83d:53045), http://dx.doi.org/10.2307/1971088
 [MY2]
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), http://dx.doi.org/10.1016/00409383(82)900210
 [MY3]
William
W. Meeks III and Shing
Tung Yau, The existence of embedded minimal surfaces and the
problem of uniqueness, Math. Z. 179 (1982),
no. 2, 151–168. MR 645492
(83j:53060), http://dx.doi.org/10.1007/BF01214308
 [Ni]
Johannes
C. C. Nitsche, Variational problems with inequalities as boundary
conditions or how to fashion a cheap hat for Giacometti’s
brother, Arch. Rational Mech. Anal. 35 (1969),
83–113. MR
0248585 (40 #1837)
 [Os]
Robert
Osserman, A proof of the regularity everywhere of the classical
solution to Plateau’s problem, Ann. of Math. (2)
91 (1970), 550–569. MR 0266070
(42 #979)
 [Ra]
Tibor
Radó, On Plateau’s problem, Ann. of Math. (2)
31 (1930), no. 3, 457–469. MR
1502955, http://dx.doi.org/10.2307/1968237
 [Ri]
David
Joseph Allyn Richardson, VARIATIONAL PROBLEMS WITH THIN
OBSTACLES, ProQuest LLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)–The
University of British Columbia (Canada). MR
2628343
 [TT]
Friedrich
Tomi and Anthony
J. Tromba, Extreme curves bound embedded minimal surfaces of the
type of the disc, Math. Z. 158 (1978), no. 2,
137–145. MR
486522 (80k:53011), http://dx.doi.org/10.1007/BF01320863
 [Al]
 H.W. Alt, Verzweigungspunkte von Flachen. II, Math. Ann. 201 (1973), 3355. MR 0331195 (48:9529)
 [AS]
 F.J. Almgren and L. Simon, Existence of embedded solutions of Plateau's problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979) no. 3, 447495. MR 553794 (81d:49025)
 [At]
 I. Athanasopoulos, Coincidence set of minimal surfaces for the thin obstacle, Manuscripta Math. 42 (1983) 199209. MR 701203 (85a:49048)
 [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)
 [CM]
 T. Colding and W.P. Minicozzi, Minimal surfaces, Courant Lecture Notes in Mathematics, 4. New York, 1999. MR 1683966 (2002b:49072)
 [Do]
 J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263321. MR 1501590
 [EWW]
 T. Ekholm, B. White, and D. Wienholtz, Embeddedness of minimal surfaces with total boundary curvature at most , Ann. of Math. (2) 155 (2002), no. 1, 209234. MR 1888799 (2003f:53010)
 [Fe]
 H. Federer, Geometric measure theory, SpringerVerlag, New York, 1969. MR 0257325 (41:1976)
 [Fr]
 J. Frehse, Two dimensional variational problems with thin obstacles, Math. Z. 143 (1975), no. 3, 279288. MR 0380550 (52:1450)
 [Ga]
 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)
 [Le]
 H. Lewy, On the coincidence set in variational inequalities, J. Differential Geometry 6 (1972) 497501. MR 0320343 (47:8882)
 [Mo]
 C.B. Morrey, The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948) 807851. MR 0027137 (10:259f)
 [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)
 [MY2]
 W. Meeks and S.T. Yau, The classical Plateau problem and the topology of three manifolds, Topology 21 (1982) 409442. MR 670745 (84g:53016)
 [MY3]
 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)
 [Ni]
 J.C.C. Nitsche, Variational problems with inequalities as boundary conditions or how to fashion a cheap hat for Giacometti's brother, Arch. Rational Mech. Anal. 35 (1969) 83113. MR 0248585 (40:1837)
 [Os]
 R. Osserman, A proof of the regularity everywhere of the classical solution to Plateau's problem, Ann. of Math. (2) 91 (1970) 550569. MR 0266070 (42:979)
 [Ra]
 T. Rado, On Plateau's problem, Ann. of Math. (2) 31 (1930) no. 3, 457469. MR 1502955
 [Ri]
 D. Richardson, Variational problems with thin obstacles, Ph.D. Thesis, The University of British Columbia (1978). MR 2628343
 [TT]
 F. Tomi and A.J. Tromba, Extreme curves bound embedded minimal surfaces of the type of the disc, Math. Z. 158 (1978) 137145. MR 486522 (80k:53011)
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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.
