New results on the classical problem of Plateau on the existence of many solutions

Reinhold Böhme

Séminaire Bourbaki (1981-1982)

  • Volume: 24, page 1-20
  • ISSN: 0303-1179

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Böhme, Reinhold. "New results on the classical problem of Plateau on the existence of many solutions." Séminaire Bourbaki 24 (1981-1982): 1-20. <http://eudml.org/doc/109987>.

@article{Böhme1981-1982,
author = {Böhme, Reinhold},
journal = {Séminaire Bourbaki},
keywords = {Riemann surface; conformal structure; conformal parametrization; Teichmüller theory},
language = {eng},
pages = {1-20},
publisher = {Société Mathématique de France},
title = {New results on the classical problem of Plateau on the existence of many solutions},
url = {http://eudml.org/doc/109987},
volume = {24},
year = {1981-1982},
}

TY - JOUR
AU - Böhme, Reinhold
TI - New results on the classical problem of Plateau on the existence of many solutions
JO - Séminaire Bourbaki
PY - 1981-1982
PB - Société Mathématique de France
VL - 24
SP - 1
EP - 20
LA - eng
KW - Riemann surface; conformal structure; conformal parametrization; Teichmüller theory
UR - http://eudml.org/doc/109987
ER -

References

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  1. 1. Alt, H.W.: Verzweigungspunkte von H-Flächen. I, II. Math. Z.127, 333 - 362, 1972, Math. Ann.201, 304 - 520, 1973. Zbl0253.58007MR312404
  2. 2. Almgren, F.J. and Simon, L.: Existence of embedded solutions of Plateau's problem. Ann. Sc. Norm. Sup. Pisa IV, 6, 447 - 495, 1979. Zbl0417.49051MR553794
  3. 3. Almgren, F.J. and Thurston, W.P.: Examples of unknotted curves which bound only surfaces of high genus within their convex hulls. 1976. Ann.Math.105,527-538, 1977 Zbl0353.53001MR442833
  4. 4. Beeson, M.: Some results on finiteness in Plateau's problem. Part. I. Math. Z.175, 103 - 123, 1980. Zbl0431.53010MR597083
  5. 5. Böhme, R.: Über Stabilität und Isoliertheit der Lösungen des Klassischen Plateauproblems. Math. Z.158, 211 - 243, 1978. Zbl0396.53004MR514127
  6. 6. Böhme, R.: A Plateau problem with many solutions. To appear in Arch. Rat.Mech.Anal.; A class of Plateau problems with many solutions. To appear in Journal f.d. Reine u. Angew. Math. 
  7. 7. Böhme, R.: The index theorem for minimal surfaces of higher genus and higher connectivity. Part I. (No boundary branch points; fixed conformal structure). Preprint. 
  8. 8. Böhme, R. and Tromba, A.J.: The index theorem for classical minimal surfaces. Ann. Math.113, 447 - 499, 1981. Zbl0482.58010MR621012
  9. 9. Böhme, R. and Tomi, F.: Zur lokalen Struktur der Lösungsmenge des Plateauproblems. Math. Z.133, 1 - 30, 1973. Zbl0273.49061MR346637
  10. 10. Courant, R.: The existence of minimal surfaces of given topological structure under prescribed boundary conditions. Acta Math.72, 51 - 98, 1940. Zbl66.0485.02MR2478JFM66.0485.02
  11. 11. Courant, R.: Dirichlet's principle, conformal mappings, and minimal surfaces. Interscience, New York, 1950. Zbl0040.34603MR36317
  12. 12. Douglas, J.: Solution of the problem of Plateau. Trans. AMS33, 263 - 321, 1931. Zbl57.1542.03MR1501590JFM57.1542.03
  13. 13. Douglas, J.: Minimal surfaces of higher topological structure. Ann. Math.40, 205 - 298, 1940. Zbl0020.37402
  14. 14. Federer, H.: Some theorems on integral currents. Trans. AMS117, 1965. Zbl0136.18204MR168727
  15. 15. Fleming, W.H.: An example in the problem of least area. Proc. AMS7, 1063 - 1074, 1956. Zbl0078.13804MR82046
  16. 16. Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. Math.97, 275 - 305, 1973. Zbl0246.53053MR317188
  17. 17. Gulliver, R. and Lesley, F.D.: On boundary branch points of minimizing surfaces. Arch. Rat. Mech. Anal.52, 1973. Zbl0263.53009MR346641
  18. 18. Gulliver, R. and Spruch, J.: On embedded minimal surfaces. Ann. Math.103, 331 - 347, 1976. Zbl0347.49033MR405217
  19. 19. Hardt, R. and Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. Math.110, 439 - 486, 1979. Bull. Amer. Math. Soc. New Series, vol. 1, 263 - 266, 1980. Zbl0421.49042
  20. 20. Heinz, E.: Über eine Verallgemeinerung des Plateau'schen Problems. manuscripta math.28, 81 - 88, 1979. Zbl0415.49027MR535696
  21. 21. Heinz, E.: Über die analytische Abhängigkeit der Lösungen eines linearen elliptischen Randwertproblems von Parametern. Nachr. Akad. Wiss. Göttingen, II., 1979, 1 - 20. Zbl0436.35010MR568799
  22. 22. Hildebrandt, St.: Boundary behavior of minimal surfaces. Arch. Rat. Mech. Anal.35, 47 - 81, 1969. Zbl0183.39402MR248650
  23. 23. Jorge, L.P. de M.: An application of the second variation of the area to the Plateau's problem. Preprint Rio de Janeiro 
  24. 24. Lawson, H.B. and Osserman, R.: Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system. Acta Math.139, 1 - 17, 1977. Zbl0376.49016MR452745
  25. 25. Lewy, H.: On the boundary behavior of minimal surfaces. Proc. Nat. Acad. Sci. USA, 37, 103 - 110, 1951. Zbl0042.15702MR49398
  26. 26. Kruskal, M.: The bridge theorem for minimal surfaces. Comm.P.Appl. Math.7(1954) 297 - 316. Zbl0055.39602MR62966
  27. 27. Marx, I.: On the classification of unstable minimal surfaces with polygonal boundaries. Comm. Pure Appl. Math.8, 235 - 244, 1955. Zbl0068.35103MR73072
  28. 28. Meeks, W.H. and Yau, S.T.: Topology of three dimensional manifolds and the embedding problem in minimal surface theory. Ann. Math.112, 441 - 484. Zbl0458.57007MR595203
  29. 29. Meeks, W.H. and Yau, S.T.: The classical Plateau problem and the topology of three dimensional manifolds. Arch. Rat. Mech. Anal. (to appear). Zbl0439.49027MR670745
  30. 30. Morgan, F.: Almost every curve in IR3 bounds a unique area minimizing surface. Inv. Math.45, 253 - 297, 1978. Zbl0378.49028MR513662
  31. 31. Morgan, F.: A smooth curve in R4 bounding a continuum of area minimizing surfaces. Duke Math. J.43, 867 - 870, 1976. Zbl0338.53003MR415475
  32. 32. Morgan, F.: Generic uniqueness for hypersurfaces minimizing the integral of an elliptic integrand with constant coefficients. Indiana Math. J. (to appear). Zbl0423.53042MR600030
  33. 33. Morgan, F.: On the singular structure of two-dimensional area minimizing surfaces in Rn. Preprint 1981. 
  34. 34. Morrey, J.C.B.: Multiple integrals and the calculus of variations. Springer, Berlin: 1964. Zbl0142.38701
  35. 35. Morse, M.: The foundations of a theory of the calculus of variations in the large in m-space (2ndpaper). Trans. Amer. Math. Soc.32, 599 - 631, 1930. Zbl56.1079.01MR1501555JFM56.1079.01
  36. 36. Morse, M. and Tompkins, C.B.: On the existence of minimal surfaces of general critical types. Proc. Nat. Acad. Sci.25, 153 - 158, Ann. Math.40, 443 - 472, 1939. Zbl0021.03404MR1503471JFM65.0456.03
  37. 37. Nitsche, J.C.C.: Contours bounding at least three solutions of Plateau's problem. Arch. Rat. Mech. Anal.30, 1 - 11, 1968. Zbl0164.14201MR226515
  38. 38. Nitsche, J.C.C.: Concerning the isolated character of solutions of Plateau's problem. Math. Z.109, 393 - 411, 1969. Zbl0189.10804MR243439
  39. 39. Nitsche, J.C.C.: A new uniqueness theorem for minimal surfaces. Arch. Rat. Mech. Anal.52, 319 - 329, 1973. Zbl0285.49001MR341258
  40. 40. Nitsche, J.C.C.: Oberwolfach Lecture1980 
  41. 41. Nitsche, J.C.C.: Vorlesungen über Minimalflächen. Springer VerlagBerlin1975. Zbl0319.53003MR448224
  42. 42. Osserman, R.: A proof of the regularity everywhere of the classical solution of Plateau's problem. Ann. Math.91, 550 - 569, 1970. Zbl0194.22302MR266070
  43. 43. Osserman, R.: The isoperimetric inequality. Bull. Amer. Math. Soc. vol. 84, 1182 - 1238, 1978. Zbl0411.52006MR500557
  44. 44. Osserman, R. and Shiffman, M.: Doubly connected minimal surfaces. Arch. Rat. Mech. Anal.58, 285 - 307, 1975. Zbl0352.53005MR385687
  45. 45. Radò, T.: The problem of least area and the problem of Plateau. Math.Z.32, 763 - 796, 1930. Zbl56.0436.01MR1545197JFM56.0436.01
  46. 46. Schüffler, K.H.: Stabilität mehrfach zusammenhängender Minimalflächen. manuscr. math.30, 163 - 197, 1979. Zbl0426.53005MR553727
  47. 47. Taylor, J.: The structure of singularities in soap-bubble-like minimal surfaces. Ann. Math.103, 489 - 539, 1976. Zbl0335.49032MR428181
  48. 48. Taylor, J.: The structure of singularities in solutions to ellipsoidal variational problems in R3. Ann. Math.103, 541 - 546, 1976. Zbl0335.49033MR428182
  49. 49. Tomi, F.: On the local uniqueness of the problem of least area. Arch. Rat. Mech. Anal.52, 312 - 318, 1973. Zbl0277.49016MR346639
  50. 50. Tomi, F.: On the finite solvability of Plateau's problem. Lecture Notes in Math. vol. 597, 679 - 695, 1977. Zbl0361.53008MR454874
  51. 51. Tomi, F. and Tromba, A.J.: Extreme curves bound an embedded minimal surface of the disctype. Math. Z.158, 1 - 13, 1978. Zbl0353.53032MR486522
  52. 52. Tomi, F. and Tromba, A.J.: On the structure of the set of curves bounding minimal surfaces of prescribed degeneracy. Journ. f.d. Reine u. Angew. Math.316, 31 - 43, 1980. Zbl0421.58002MR581321
  53. 53. Tromba, A.J.: A general approach to Morse theory. Journal of Diff. Geom.12, 47 - 85, 1977. Zbl0344.58012MR464304
  54. 54. Tromba, A.J.: The set of curves of uniqueness for Plateau's problem has a dense interior. Lect. Notes Math. vol. 597, 696 - 707, 1977. Zbl0361.53007MR451294
  55. 55. Tromba, A.J.: On the number of simply connected minimal surfaces spanning a curve.Memoirs AMS12, No 194, 1977. Zbl0389.53003MR649458
  56. 56. Tromba, A.J.: Degree Theory on Oriented Infinite Dimensional Varieties and the Morse Number of Minimal Surfaces Spanning a Curve in Rn. Preprint: Bonn1981. 
  57. 57. Tromba, A.J.A differential geometric approach to Teichmüller theory and the smoothness of variations of the conformal structure. Lectures1980, to appear SFB72, Bonn1981. 
  58. 58. Wente, H.C.: The Plateau problem for symmetric surfaces. Arch. Rat. Mech. Anal.60, 149 - 169, 1976. Zbl0328.53003MR420448
  59. 59. Morse, M. and Tompkins, C.: Unstable minimal surfaces of higher topological structure. Duke Math. J.8, 350 - 375, 1941. Zbl0025.40902MR4744JFM67.1040.02
  60. 60. Shiffman, M.: The Plateau problem for non-relative minima. Ann. of Math.40, 834 - 854, 1939. Zbl0023.39802MR467
  61. 61. Shiffman, M.: Unstable minimal surfaces with several boundaries. Ann. of Math.43, 197 - 222, 1942. Zbl0063.06961MR6030

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