Skip to main content
SpringerLink
Log in
Book cover

Geometry V pp 5–93Cite as

Complete Embedded Minimal Surfaces of Finite Total Curvature

  • Chapter

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 90))

Abstract

We will survey what is known about minimal surfaces S ⊂ R 3, which are complete, embedded, and have finite total curvature: \(\int_s {|K|} dA < \infty \). The only classically known examples of such surfaces were the plane and the catenoid. The discovery by Costa [14, 15] early in the last decade, of a new example that proved to be embedded sparked a great deal of research in this area. Many new examples have been found, even families of them, as will be described below. The central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples.

Supported by research grant DE-FG03-95ER25250 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, and National Science Foundation, Division of Mathematical Sciences research grant DMS-9596201. Research at MSRI is supported in part by NSF grant DMS-90-22140.

Partially supported by Sonderforschungsbereich SFB256 at Bonn.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barbosa, J.L., and do Carmo, M.: On the size of a stable minimal surface in R 3. Am. J. Math. 98 (2), 515–528 (1976), Zbl. 332.53006

    Article  MATH  Google Scholar 

  2. Barbosa, L., and Colares, A.G.: Minimal surfaces in R3. Lecture Notes Math. 1195. Springer-Verlag 1986, Zbl. 609.53001

    Google Scholar 

  3. Bloß, D.: Elliptische Funktionen und vollständige Minimalflächen. PhD thesis, FU Berlin, 1989, Zbl. 708.53007

    Google Scholar 

  4. Boix, E., and Capdequi, L.: Deformation families of complete minimal surfaces with four catenoid ends. (Video tape) GANG, August 1991

    Google Scholar 

  5. Boix, E., and Wohlgemuth, M.: Numerical results on embedded minimal surfaces of finite total curvature. In preparation

    Google Scholar 

  6. Bryant, R.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20, 23–53 (1984), Zbl. 555.53002

    MATH  Google Scholar 

  7. Bryant, R.: Surfaces in conformal geometry. In: The mathematical heritage of Hermann Weyl, Proc. Symp. Durham/NC 1987, Proc. Symp. Pure Math. 48, 227–240 (1988), Zbl. 654.53010

    Google Scholar 

  8. Callahan, M., Hoffman, D., and Hoffman, J.: Computer graphics tools for the study of minimal surfaces. Commun. ACM 31 (6), 648–661 (1988)

    Article  MathSciNet  Google Scholar 

  9. Callahan, M., Hoffman, D., and Meeks III, W.H.: Embedded minimal surfaces with an infinite number of ends. Invent. Math. 96, 459–505 (1989), Zbl. 676.53004

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, C.C., and Gackstatter, F.: Elliptic and hyperelliptic functions and complete minimal surfaces with handles. Instituto de Matemática e Estatistica Universidade de São Paulo 27, 1981

    Google Scholar 

  11. Chen, C.C., and Gackstatter, F.: Elliptische und hyperelliptische Funktionen und vollständige Minimalflächen vom Enneperschen Typ. Math. Ann. 259, 359–369 (1982), Zbl. 468.53008

    Article  MathSciNet  MATH  Google Scholar 

  12. Choe, J.: Index, vision number, and stability of complete minimal surfaces. Arch. Ration. Mech. Anal. 109 (3), 195–212 (1990), Zbl. 695.53045

    Article  MathSciNet  MATH  Google Scholar 

  13. Choi, T., Meeks III, W.H., and White, B.: A rigidity theorem for properly embedded minimal surfaces in R 3. J. Differ. Geom. 32, No. 1, 65–76 (1990), Zbl. 704.53008

    MathSciNet  MATH  Google Scholar 

  14. Costa, C.: Imersöes minimas en R3 de gênero un e curvatura total finita. PhD thesis, IMPA, Rio de Janeiro, Brasil 1982

    Google Scholar 

  15. Costa, C.: Example of a complete minimal immersion in R3 of genus one and three embedded ends. Bol. Soc. Bras. Mat. 15, 47–54 (1984), Zbl. 613.53002

    Article  MATH  Google Scholar 

  16. Costa, C.: Uniqueness of minimal surfaces embedded in R3 with total curvature 12π. J. Differ. Geom. 30 (3), 597–618 (1989), Zbl. 696.53001

    MATH  Google Scholar 

  17. Dierkes, U., Hildebrandt, S., Küster, A., and Wohlrab, O.: Minimal Surfaces I. Grundl. math. Wiss. 295. Springer-Verlag 1992, Zbl. 777.53012

    Book  Google Scholar 

  18. do Carmo, M., and Peng, C.K.: Stable minimal surfaces in R 3 are planes. Bull. Am. Math. Soc., New Ser. 1, 903–906 (1979), Zbl. 442.53013

    Article  MATH  Google Scholar 

  19. do Espírito-Santo, N.: SuperfIcies mínimas completas em R 3 com fim de tipo Enneper. PhD thesis, University of Niteroi, Brazil 1992. [See also Ann. Inst. Fourier 44, No. 2, 525–577 (1994), Zbl. 803.53006]

    Google Scholar 

  20. Ejiri, N., and Kotani, M.: Minimal surfaces in S2m(1) with extra eigenfunctions. Q. J. Math. Oxf., II. Ser. 172 421–440 (1992), Zbl. 778.53047

    Article  MathSciNet  Google Scholar 

  21. Fang, Y., and Meeks III, W.H.: Some global properties of complete minimal surfaces of finite topology in R3. Topology 30 (1), 9–20 (1991), Zbl. 737.53010

    Article  MathSciNet  MATH  Google Scholar 

  22. Fischer-Colbrie, D.: On complete minimal surfaces with finite Morse index in 3-manifolds. Invent. Math. 82, 121–132 (1985), Zbl. 573.53038

    Article  MathSciNet  MATH  Google Scholar 

  23. Fischer-Colbrie, D., and Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature. Comm. Pure Appl. Math. 33, 199–211 (1980), Zbl. 439.53060

    Article  MathSciNet  MATH  Google Scholar 

  24. Gackstatter, F.: Uber die Dimension einer Minimalfläche und zur Ungleichung von St. Cohn-Vossen. Arch. Ration. Mech. Anal. 61 (2), 141–152 (1976), Zbl. 328.53002

    Article  MathSciNet  MATH  Google Scholar 

  25. Gulliver, R.: Index and total curvature of complete minimal surfaces. Proc. Symp. Pure Math. 44 207–212 (1986), Zbl. 592.53006

    Article  MathSciNet  Google Scholar 

  26. Gulliver, R., and Lawson, H.B.: The structure of minimal hypersurfaces near a singularity. Proc. Symp. Pure Math. 44 213–237 (1986), Zbl. 592.53005

    Article  MathSciNet  Google Scholar 

  27. Hadamard, J.-J.: Les surfaces à courbures opposées et leurs lignes géodésiques. J. Math. Pures Appl. 4, 27–73 (1898)

    Google Scholar 

  28. Hoffman, D.: Embedded minimal surfaces computer graphics and elliptic functions. In: Proceedings of the Berlin Conference on Global Differential Geometry, Lecture Notes Math. 1156, 204–215. Springer-Verlag 1985, Zbl. 566.53016

    Google Scholar 

  29. Hoffman, D.: The computer-aided discovery of new embedded minimal surfaces. Math. Intell. 9 (3), 8–21 (1987), Zbl. 616.53007

    Article  MATH  Google Scholar 

  30. Hoffman, D., and Meeks III, W.H.: One-parameter families of embedded complete minimal surfaces of finite topology. In preparation

    Google Scholar 

  31. Hoffman, D., Karcher, H., and Rosenberg, H.: Embedded minimal annuli in R3 bounded by a pair of straight lines. Comment. Math. Helvet. 66, 599–617 (1991), Zbl. 765.53004

    Article  MathSciNet  MATH  Google Scholar 

  32. Hoffman, D., Karcher, H., and Wei, F.: Adding handles to the helicoid. Bull. Am. Math. Soc., New Ser. 29 (1), 77–84 (1993), Zbl. 787.53003

    Article  MathSciNet  MATH  Google Scholar 

  33. Hoffman, D., Karcher, H., and Wei, F.: The genus one helicoid and the minimal surfaces that led to its discovery. In: Global Analysis and Modern Mathematics. Publish or Perish Press 1993. K. Uhlenbeck (ed.), 119–170

    Google Scholar 

  34. Hoffman, D., and Meeks III, W.H.: A variational approach to the existence of complete embedded minimal surfaces. Duke Math. J. 57 (3), 877–893 (1988), Zbl. 676.53006

    Article  MathSciNet  MATH  Google Scholar 

  35. Hoffman, D., and Meeks III, W.H.: The asymptotic behavior of properly embedded minimal surfaces of finite topology. J. Am. Math. 2 (4), 667–682 (1989), Zbl. 683.53005

    Article  MathSciNet  MATH  Google Scholar 

  36. Hoffman, D., and Meeks III, W.H.: Embedded minimal surfaces of finite topology. Ann. Math., II. Ser. 131, 1–34 (1990), Zbl. 695.53004

    Article  MathSciNet  MATH  Google Scholar 

  37. Hoffman, D., and Meeks III, W.H.: Les surfaces minimales: la caténoide par les deux bouts. Quadrature 5, 31–47 (1990)

    Google Scholar 

  38. Hoffman, D., and Meeks III, W.H.: Limits of minimal surfaces and Scherk’s Fifth Surface. Arch. Ration. Mech. Anal. 111 (2), 181–195 (1990), Zbl. 709.53006

    Article  MathSciNet  MATH  Google Scholar 

  39. Hoffman, D., and Meeks III, W.H.: Minimal surfaces based on the catenoid. Am. Math. Mon. 97 (8), 702–730 (1990), Zbl. 737.53006

    Article  MathSciNet  MATH  Google Scholar 

  40. Hoffman, D., and Meeks III, W.H.: The strong halfspace theorem for minimal surfaces. Invent. Math. 101, 373–377 (1990), Zbl. 719.53037

    Article  MathSciNet  MATH  Google Scholar 

  41. Hoffman, J.T.: MESH manual. GANG preprint series II, #35

    Google Scholar 

  42. Jorge, L., and Meeks III, W.H.: The topology of complete minimal surfaces of finite total Gaussian curvature. Topology 22 (2), 203–221 (1983), Zbl. 517.53008

    Article  MathSciNet  MATH  Google Scholar 

  43. Karcher, H.: Construction of minimal surfaces. Surveys in Geometry, 1–96, 1989. University of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989

    Google Scholar 

  44. Kusner, R.: Conformal geometry and complete minimal surfaces. Bull. Am. Math. Soc. 17 (2), 291–295 (1987), Zbl. 634.53004

    Article  MathSciNet  MATH  Google Scholar 

  45. Kusner, R.: Global geometry of extremal surfaces in three-space. PhD thesis, University of California, Berkeley, 1988

    Google Scholar 

  46. Kusner, R.: Bubbles, conservation laws and balanced diagrams. In: Geometric Analysis and Computer Graphics, volume 17 of MSRI Publications. SpringerVerlag, Berlin Heidelberg New York 1990. P. Concus and R. Finn and D. Hoffman (eds.), Zbl. 776.68009

    Google Scholar 

  47. Lawson, H.B.: Complete minimal surfaces in S3. Ann. Math., II. Ser. 92, 335– 374 (1970)

    Google Scholar 

  48. Lawson, Jr. H.B.: Lectures on Minimal Submanifolds. Publish or Perish Press, Berkeley, 1971, Zbl. 434.53006

    Google Scholar 

  49. Lopez, F.J.: The classification of complete minimal surfaces with total curvature greater than —12π. Trans. Am. Math. Soc. 334 (1), 49–74 (1992), Zbl. 771.53005

    MATH  Google Scholar 

  50. Lopez, F.J., and Ros, A.: Complete minimal surfaces with index one and stable constant mean curvature surfaces. Comment. Math. Helvet. 64, 34–43 (1989), Zbl. 679.53047

    Article  MathSciNet  MATH  Google Scholar 

  51. Lopez, F.J., and Ros, A.: On embedded complete minimal surfaces of genus zero. J. Differ. Geom. 33 (1), 293–300 (1991), Zbl. 719.53004

    MathSciNet  MATH  Google Scholar 

  52. Meeks III, W.H., and Rosenberg, H.: The maximum principle at infinity for minimal surfaces in f;at three-manifolds. Comment. Math. Helvet. 65, 255–270 (1990), Zbl. 713.53008

    Article  MathSciNet  MATH  Google Scholar 

  53. Meeks III, W.H., and Rosenberg, H.: The geometry and conformal structure of properly embedded minimal surfaces of finite topology in R3. Invent. Math. 114, 625–639 (1993), Zbl. 803.53007

    Article  MathSciNet  MATH  Google Scholar 

  54. Montiel, S., and Ros, A.: Schrödinger operators associated to a holomorphic map. In: Global Differential Geometry and Global Analysis (Berlin, 1990), Proc. Conf. Berlin/Ger. 1990, Lecture Note Math. 1481, 147–174. Springer-Verlag, Berlin Heidelberg New York 1990, Zbl. 744.58007

    Google Scholar 

  55. Nayatani, S.: On the Morse index of complete minimal surfaces in Euclidean space. Osaka J. Math. 27, 441–451 (1990), Zbl. 704.53007

    MathSciNet  MATH  Google Scholar 

  56. Nayatani, S.: Morse index of complete minimal surfaces. In: The Problem of Plateau, 181–189. World Scientific Press, 1992. Th. M. Rassias (ed.), Zbl. 794.58011

    Google Scholar 

  57. Nitsche, J.C.C.: A characterization of the catenoid. J. Math. Mech. 11, 293–301 (1962), Zbl. 106, 146 58.

    MathSciNet  MATH  Google Scholar 

  58. Osserman, R.: Global properties of minimal surfaces in E3 and En. Ann. Math., II. Ser. 80 (2), 340–364 (1964), Zbl. 134, 385

    Article  MathSciNet  MATH  Google Scholar 

  59. Osserman, R.: A Survey of Minimal Surfaces. Dover Publications, New York, 2nd edition, 1986. (lst ed. Van Nostrand, New York 1969, Zbl. 209, 529)

    Google Scholar 

  60. Pérez, J., and Ros, A.: Some uniqueness and nonexistence theorems for embedded minimal surfaces. Math. Ann. 295 (3), 513–525 (1993), Zbl. 789.53004

    Article  MathSciNet  MATH  Google Scholar 

  61. Pogorelov, A.V.: On the stability of minimal surfaces. Dokl. Akad. Nauk SSSR 260, 293–295 (1981, Zbl. 495.53005) English transl.: Soviet Math. Dokl. 24, 274–276 (1981)

    MathSciNet  Google Scholar 

  62. Riemann, B.: Uber die Fläche vom kleinsten Inhalt bei gegebener Begrenzung. Abh. Königl, d. Wiss. Göttingen, Mathem. Cl. 13, 3–52 (1867)

    Google Scholar 

  63. Riemann, B.: Ouevres Mathématiques de Riemann. Gauthiers-Villars, Paris 1898, Jbuch 29, 9

    Google Scholar 

  64. Rosenberg, H.: Deformations of minimal surfaces. Trans. Am. Math. Soc. 295, 475–290 (1986)

    Article  MATH  Google Scholar 

  65. Rosenberg, H., and Toubiana, E.: Some remarks on deformations of minimal surfaces. Trans. Am. Math. Soc. 295, 491–499 (1986), Zbl. 598.53005

    Article  MathSciNet  MATH  Google Scholar 

  66. Rosenberg, H., and Toubiana, E.: A cylindrical type complete minimal surface in a slab of R 3. Bull. Sci. Math., II. Ser. 111, 241–245 (1987), Zbl. 631.53012

    MathSciNet  MATH  Google Scholar 

  67. Schoen, R.: Uniqueness, symmetry, and embeddedness of minimal surfaces. J. Differ. Geom. 18, 791–809 (1983), Zbl. 575.53037

    MathSciNet  MATH  Google Scholar 

  68. Schwarz, H.A.: Gesammelte Mathematische Abhandlungen, volume 1. SpringerVerlag, Berlin 1890, Jbuch 22, 31

    Book  MATH  Google Scholar 

  69. Souam, R.: Exemples d’applications holomorphes d’indice un. Ann. Inst. Fourier 43 (2), 369–381 (1993), Zbl. 788.30037

    Article  MathSciNet  MATH  Google Scholar 

  70. Thayer, E.: Higher-genus Chen-Gackstatter minimal surfaces, Scherk’s singlyperiodic minimal surface and Riemann surfaces of infinite genus. Experimental Math. 4, No. 1, 19–39 (1995)

    Article  MathSciNet  Google Scholar 

  71. Thayer, E.: Complete Minimal Surfaces in Euclidean 3-Space. PhD thesis, University of Massachusetts at Amherst, 1994

    Google Scholar 

  72. Tysk, J.: Eigenvalue estimates with applications to minimal surfaces. Pac. J. Math. 128, 361–366 (1987), Zbl. 632.58016

    Article  MathSciNet  MATH  Google Scholar 

  73. Wohlgemuth, M.: Higher genus minimal surfaces by growing handles out of a catenoid. Manuscr. Math. 70, 397–428 (1991), Zbl. 752.53008

    Article  MathSciNet  MATH  Google Scholar 

  74. Wohlgemuth, M.: Vollständige Minimalflächen höheren Geschlechts und endlicher Totalkrümmung. PhD thesis, Bonner Mathematische Schriften 262, University of Bonn, April 1993

    Google Scholar 

  75. Wohlgemuth, M.: Higher genus minimal surfaces of finite total curvature. Arch. Ration. Mech. Anal. (to appear)

    Google Scholar 

  76. Yang, K.: Complete minimal surfaces of finite total curvature. Kluwer Acad. Pub., Dordrecht, Netherlands 1994

    Book  MATH  Google Scholar 

Download references

Authors
  1. David Hoffman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hermann Karcher
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA, 94720, USA

    R. Osserman (Deputy Director) (Deputy Director)

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Hoffman, D., Karcher, H. (1997). Complete Embedded Minimal Surfaces of Finite Total Curvature. In: Osserman, R. (eds) Geometry V. Encyclopaedia of Mathematical Sciences, vol 90. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03484-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-03484-2_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08225-2

  • Online ISBN: 978-3-662-03484-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation