Some recent developments in the theory of properly embedded minimal surfaces in 3

Harold Rosenberg

Séminaire Bourbaki (1991-1992)

  • Volume: 34, page 463-535
  • ISSN: 0303-1179

How to cite

top

Rosenberg, Harold. "Some recent developments in the theory of properly embedded minimal surfaces in $\mathbb {R}^3$." Séminaire Bourbaki 34 (1991-1992): 463-535. <http://eudml.org/doc/110163>.

@article{Rosenberg1991-1992,
author = {Rosenberg, Harold},
journal = {Séminaire Bourbaki},
keywords = {stable minimal surfaces; annular end theorem; periodic minimal surfaces; finite total curvature},
language = {eng},
pages = {463-535},
publisher = {Société Mathématique de France},
title = {Some recent developments in the theory of properly embedded minimal surfaces in $\mathbb \{R\}^3$},
url = {http://eudml.org/doc/110163},
volume = {34},
year = {1991-1992},
}

TY - JOUR
AU - Rosenberg, Harold
TI - Some recent developments in the theory of properly embedded minimal surfaces in $\mathbb {R}^3$
JO - Séminaire Bourbaki
PY - 1991-1992
PB - Société Mathématique de France
VL - 34
SP - 463
EP - 535
LA - eng
KW - stable minimal surfaces; annular end theorem; periodic minimal surfaces; finite total curvature
UR - http://eudml.org/doc/110163
ER -

References

top
  1. [B.Do C.] J.L. Barbosa and M. Do Carmo. On the size of a stable minimal surface in IR3. American Journal of Mathematics98(2) : 515-528, 1976. Zbl0332.53006MR413172
  2. [C.-H.-M] M. Callahan, D. Hoffman, and W.H. Meeks III. The structure of singlyperiodic minimal surfaces. Inventiones Math.99 : 455-481, 1990. Zbl0695.53005MR1032877
  3. [Cost.-1] C. Costa. Imersöes minimas en IR3 de gênero un e curvatura total finita. PhD thesis, IMPA, Rio de Janeiro, Brazil, 1982. 
  4. [Cost.-2] C. Costa. Example of a complete minimal immersion in IR3 of genus one and three embedded ends. Bull. Soc. Bras. Mat.15 : 47-54, 1984. Zbl0613.53002MR794728
  5. [Cost.-3] C. Costa. Uniqueness of minimal surfaces embedded in IR3 with total curvature -12π. Journal of Differential Geometry30(3) : 597-618, 1989. Zbl0696.53001
  6. [Cour.] R. Courant. Dirichlet's Principle, Conformal Mapping and Minimal Surfaces. Interscience Publishers, Inc., New York, 1950. Zbl0040.34603MR36317
  7. [Darb.] G. Darboux. Leçons sur la théorie générale des surfaces et les applications géometriques du calcul infinitésimal. Gauthier-Villars, Paris, 1st part, 2nd edition, 1914. JFM45.0881.05
  8. [Do C.-P.] M. Do Carmo and C.K. Peng. Stable minimal surfaces in IR3 are planes. Bulletin of the AMS1 : 903-906, 1979. Zbl0442.53013MR546314
  9. [Doug.] J. Douglas, Solution of the problem of Plateau, Trans. AMS33 : 263-321, 1931. Zbl57.1542.03MR1501590JFM57.1542.03
  10. [F.-Oss.] R. Finn and R. Osserman. On the Gauss curvature of non-parametric minimal surfaces, J. Anal. Math.12 : 351-364, 1964. Zbl0122.16404MR166694
  11. [F.C.] D. Fischer-Colbrie. On complete minimal surfaces with finite Morse index in 3-manifolds. Inventiones Math.82 : 121-132, 1985. Zbl0573.53038MR808112
  12. [Fr.-M.] C. Frohman and W.H. Meeks III. The topological uniqueness of complete one-ended minimal surfaces and Heegard surfaces in IR3, preprint. Zbl0886.57015
  13. [Fuj.-1] H. Fujimoto. On the number of exceptional values of the Gauss maps of minimal surfaces. Journal of the Math. Society of Japan40(2) : 235-247, 1988. Zbl0629.53011MR930599
  14. [Fuj.-2] H. Fujimoto. Modified defect relations for the Gauss map of minimal surfaces. Journal of Differential Geometry29 : 245-262, 1989. Zbl0676.53005MR982173
  15. [G.-T.] D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of Zbl1042.35002
  16. second order. Springer-Verlag, New York, 2nd edition, 1983. 
  17. [H.-S.] R. Hardt and L. Simon. Boundary reguarity and embedded minimal solutions for the oriented Plateau problem. Annals of Math. 110 : 439- 486, 1979. Zbl0457.49029MR554379
  18. [Heinz] E. Heinz. Über die Lösungen der Minimalflächengleichung. Nachr. Akad. Wiss. Göttingen Math. Phys. K1, II (1952) 51-56. Zbl0048.15401MR54182
  19. [H.-M.-1] D. Hoffman and W.H. Meeks III. A complete embedded minimal surface in IR3 with genus one and three ends. Journal of Differential Geometry21 : 109-127, 1985. Zbl0604.53002MR806705
  20. [H.-M.-2] D. Hoffman and W.H. Meeks III. Properties of properly embedded minimal surfaces of finite total curvature. Bulletin of the AMS17(2) : 296-300, 1987. Zbl0634.53003MR903736
  21. [H.-M.-3] D. Hoffman and W.H. Meeks III. The asymptotic behavior of properly embedded minimal surfaces of finite topology. Journal of AMS2(4) : 667-681, 1989 Zbl0683.53005MR1002088
  22. [H.-M.-4] D. Hoffman and W.H. Meeks III. The strong halfspace theorem for minimal surfaces. Inventiones Math.101 : 373-377, 1990. Zbl0722.53054MR1062966
  23. [H.-M.-5] D. Hoffman and W.H. Meeks III. Minimal surfaces based on the catenoid. Amer. Math. Monthly, Special Geometry Issue97(8) : 702-730, 1990. Zbl0737.53006MR1072813
  24. [H.-Wei] D. Hoffman and F. Wei. Adding handles to the helicoid, preprint. Zbl0787.53003
  25. [E.H.] E. Hopf. On an inequality for minimal surfaces z = f(x, y), J. Rat. Mech. Anal.2 : 519-522, 1953. Zbl0051.12601MR55735
  26. [Hub.] A. Huber. On subharmonic functions and differential geometry in the large. Commentari Mathematici Helvetici32 : 181-206, 1957. Zbl0080.15001MR94452
  27. [J.-S.] H. Jenkins, J. Serrin. Variational problems of minimal surface type II, Arch. Rat. Mech. Analysis21 : 321-342, 1966. Zbl0171.08301MR190811
  28. [J.-Xav.] L. Jorge, F. Xavier. A complete minimal surface in a slab of IR3, Annals of Maths, 1980, 203-206. Zbl0455.53004MR584079
  29. [K.-1] H. Karcher. Construction of minimal surfaces. Surveys in Geometry, pages 1-96, 1989. University of Tokyo, 1989, and Lecture Notes No.12, SFB256, Bonn, 1989. 
  30. [K.-2] H. Karcher. Embedded minimal surfaces derived from Scherk's examples. Manuscripta Math.62 : 83-114,1988. Zbl0658.53006MR958255
  31. [K.-3] H. Karcher. The triply periodic minimal surfaces of Alan Schoen and Zbl0687.53010
  32. their constant mean curvature companions. Manuscripta Math.64 : 291- 357, 1989. Zbl0687.53010MR1003093
  33. [K.-4] H. Karcher. Construction of higher genus embedded minimal surfaces. Geom. and Top. of Sub. III World Sc.174-191, 1990. Zbl0737.53007MR1344467
  34. [L.-R.] R. Langevin and H. Rosenberg. A maximum principle at infinity for minimal surfaces and applications. Duke Math. Journal57 : 819-828, 1988. Zbl0667.49024MR975123
  35. [Lo.-Ros] F.J. Lopez and A. Ros. On embedded complete minimal surfaces of genus zero. Journal of Differential Geometry33(1) : 293-300, 1991. Zbl0719.53004MR1085145
  36. [M.-1] W.H. Meeks III. The geometry, topology and existence of periodic minimal surfaces, preprint. 
  37. [M.-2] W.H. Meeks III. Lectures on Plateau's Problem. Insituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil, 1978. 
  38. [M.-3] W.H. Meeks III. The theory of triply-periodic minimal surfaces. Indiana University Math. Journal39(3) : 877-936, 1990. Zbl0721.53057MR1078743
  39. [M.-R.-1] W.H. Meeks III and H. Rosenberg. The global theory of doubly periodic minimal surfaces. Inventiones Math.97 : 351-379, 1989. Zbl0676.53068MR1001845
  40. [M.-R.-2] W.H. Meeks III and H. Rosenberg. The maximum principle at infinity for minimal surfaces in flat three-manifolds. Commentari Mathematici Helvetici65 : 255-270, 1990. Zbl0713.53008MR1057243
  41. [M.-R.-3] W.H. Meeks III and H. Rosenberg. The geometry and conformal structure of properly embedded minimal surfaces of finite topology in IR3, to appear in Invent. Math. Zbl0803.53007
  42. [M.-R.-4] W.H. Meeks III and H. Rosenberg. The geometry of periodic minimal surfaces, to appear in Comment. Math. Helv. Zbl0807.53049MR1241472
  43. [M.-Wh.] W.H. Meeks III and B. White. Minimal surfaces bounded by convex curves in parallel planes. Commentari Mathematici Helvetici66 : 263- 278, 1991. Zbl0731.53004MR1107841
  44. [M.-Y.] W.H. Meeks and S.T. Yau. The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z.179 : 151-168, 1982. Zbl0479.49026MR645492
  45. [N.] J.C.C. Nitsche. A characterization of the catenoid. Journal of Math. Mech.11 : 293-302, 1962. Zbl0106.14602MR137043
  46. [Oss.-1] R. Osserman. Global properties of minimal surfaces in E3 and En. Annals of Math.80(2) : 340-364, 1964. Zbl0134.38502MR179701
  47. [Oss.-2] R. Osserman. On the Gauss curvature of minimal surfaces. Trans. AMS96 : 115-128, 1960. Zbl0093.34303MR121723
  48. [P.-Ros] J. Pérez and A. Ros. Some uniqueness and nonexistence theorems for embedded minimal surfaces, preprint. Zbl0789.53004MR1204835
  49. [Rado-1] T. Rado. The problem of the least area and the problem of Plateau. Math. Z.32 : 763-796, 1930. MR1545197JFM56.0436.01
  50. [Rado-2] T. Rado. On the problem of Plateau. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin1933. Zbl59.1341.01MR344979JFM59.1341.01
  51. [Reif.] R. Reifenberg. Solution for the Plateau problem for m-dimensional surfaces of varying topological type. Acta Math.104 : 1-92, 1960. Zbl0099.08503MR114145
  52. [R.-T.-1] H. Rosenberg and E. Toubiana. A cylindrical type complete minimal surface in a slab of IR3. Bull. Sc. Math. III, pages 241-245, 1987. Zbl0631.53012MR912952
  53. [R.-T.-2] H. Rosenberg and E. Toubiana. Complete minimal surfaces and minimal herissons. Journal of Differential Geometry28 : 115-132, 1988. Zbl0657.53004MR950557
  54. [R.-S.E.] Sa Earp and H. Rosenberg. The Dirichlet problem for the minimal surface equation on unbounded planar domains. Journal de Mathématiques Pures et Appliquées68 : 163-183, 1989. Zbl0696.49069MR1010767
  55. [Sch.-1] R. Schoen. Uniqueness, symmetry, and embeddedness of minimal surfaces. Journal of Differential Geometry18 : 791-809, 1983. Zbl0575.53037MR730928
  56. [Sch.-2] R. Schoen. Estimates for Stable Minimal Surfaces in Three Dimensional Manifolds, volume 103 of Annals of Math. Studies. Princeton University Press, 1983. Zbl0532.53042MR795231
  57. [Simon] L. Simon. Lectures on geometric measure theory. In Proceedings of the Center for Mathematical Analysis, volume 3, Canberra, Australia, 1983. Australian National University. Zbl0546.49019MR756417
  58. [Smale] N. Smale. A bridge principle for minimal and constant mean curvature submanifolds of IRn. Invent. Math. 90 : 505-549, 1987. Zbl0637.49020MR914848
  59. [M.S.] M. Soret. Deformations de surfaces minimales. Thèse Univ. ParisVII, 1992. 
  60. [Souam] R. Souam. Stabilité et unicité des surfaces minimales. Thèse Univ. ParisVII, 1992. Zbl0880.53009
  61. [T.] E. Toubiana. On the uniqueness of the helicoid. Ann. Inst. Four.38 : 121-132, 1988. Zbl0645.53032MR978243
  62. [Wei] F. Wei. Some existence and uniqueness theorems for doubly periodicminimal surfaces, to appear in Invent. Math. Zbl0773.53005MR1168368
  63. [Wh.] B. White. Complete surfaces of finite total curvature. Journ. Diff. Geom.26 : 315-326, 1987. Zbl0631.53007MR906393

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.