Skip to main content
SpringerLink
Log in

A Brief Historical Introduction and Motivations

  • Chapter
Constant Mean Curvature Surfaces with Boundary

Part of the book series: Springer Monographs in Mathematics ((SMM))

  • 1763 Accesses

Abstract

We do a historical review of the surfaces with constant mean curvature motivating its study by the classical isoperimetric problem. We describe the two ways that historically have gone parallel, depending if in the variational characterization we consider only one constraint (the volume) or two constrains (the volume and the boundary). In the first case, we recall the different characterizations of the sphere in the family of closed surfaces with constant mean curvature. The absence of new examples brought as a consequence a great effort in the seek of new surfaces and techniques that succeeded in the discovery by Wente of a torus immersed in Euclidean space with constant mean curvature. This opened news sights of the theory. For surfaces with boundary, Douglas and Radó considered the Plateau problem for minimal surfaces and later, in the general case of mean curvature, by Heinz, Hildebrandt and others in the fifties of XX century.

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

Access this chapter

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 EPUB and 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

Institutional subscriptions

References

  1. Abresch, U.: Constant mean curvature tori in terms of elliptic functions. J. Reine Angew. Math. 394, 169–192 (1987)

    MathSciNet  Google Scholar 

  2. Adamson, A.W., Gast, A.P.: Physical Chemistry of Surfaces, 4th edn. Wiley, New York (1982)

    Google Scholar 

  3. Alexandrov, A.D.: Uniqueness theorems for surfaces in the large I–V. Vestn. Leningr. Univ. 11 #19, 5–17 (1956); 12 #7, 15–44, 1957; 13 #7, 14–26, 1958; 13 #13, 27–34, 1958; 13 #19, 5–8, (1958). English transl. in Amer. Math. Soc. Transl. 21, 341–354, 354–388, 389–403, 403–411, 412–416 (1962)

    Google Scholar 

  4. Alexandrov, A.D.: A characteristic property of spheres. Ann. Mat. Pura Appl. 58, 303–315 (1962)

    Article  MathSciNet  Google Scholar 

  5. Almgren, F.J.: Plateau’s Problem: An Invitation to Varifold Geometry, 1st edn. Mathematics Monographs Series. Benjamin, New York (1966)

    MATH  Google Scholar 

  6. Andrews, B., Li, H.: Embedded constant mean curvature tori in the three-sphere. arXiv:1204.5007v3 (2012)

  7. Barbosa, J.L., do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Z. 185, 339–353 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Barbosa, J.L., Colares, A.G.: Minimal Surfaces in \(\mathbb{R}^{3}\). Lecture Notes in Math., vol. 1195. Springer, Berlin (1986)

    Google Scholar 

  9. Bernstein, S.N.: Sur la généralization du problème de Dirichlet, II. Math. Ann. 69, 82–136 (1910)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bethuel, F., Caldiroli, P., Guida, M.: Parametric surfaces with prescribed mean curvature 2001. Rend. Semin. Mat. (Torino) 60, 175–231 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Bobenko, A.I.: All constant mean curvature tori in \(\mathbb{R}^{3}\), \(\mathbb{S}^{3}\), \(\mathbb{H}^{3}\) in terms of theta functions. Math. Ann. 290, 209–245 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Boys, C.V.: Soap Bubbles: Their Colors and Forces Which Mold Them. Dover, New York (1959)

    Google Scholar 

  13. Breiner, C., Kapouleas, N.: Embedded constant mean curvature surfaces in Euclidean three space (2012). arXiv:1210.3394

  14. Brendle, S.: Embedded minimal tori in S 3 and the Lawson conjecture. To appear Acta Math.

    Google Scholar 

  15. Brézis, H., Coron, J.M.: Sur la conjecture de Rellich pour les surfaces à courbure moyenne prescrite. C. R. Acad. Sci. Paris Sér. I Math. 295, 615–618 (1982)

    MATH  Google Scholar 

  16. Brézis, H., Coron, J.M.: Multiple solutions of H-systems and Rellich’s conjecture. Commun. Pure Appl. Math. 37, 149–187 (1984)

    Article  MATH  Google Scholar 

  17. Colding, T., Minicozzi, W.: In: Minimal Surfaces. Courant Lect. Notes Math., vol. 4, New York (1999)

    Google Scholar 

  18. Colding, T., Minicozzi, W.: A Course in Minimal Surfaces. Graduate Studies in Mathematics, vol. 121. Am. Math. Soc., Providence (2011)

    MATH  Google Scholar 

  19. Courant, R.: Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces. Interscience, New York (1950)

    MATH  Google Scholar 

  20. Delaunay, C.: Sur la surface de révolution dont la courbure moyenne est constante. J. Math. Pures Appl. 6, 309–315 (1841)

    Google Scholar 

  21. Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.: Minimal Surfaces I, II. Springer, Berlin (1992)

    Book  Google Scholar 

  22. Dorfmeister, J., Haak, G.: On constant mean curvature surfaces with periodic metric. Pac. J. Math. 182, 229–287 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dorfmeister, J., Haak, G.: Investigation and application of the dressing action on surfaces of constant mean curvature. Q. J. Math. 51, 57–73 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dorfmeister, J., Haak, G.: Construction of non-simply-connected CMC surfaces via dressing. J. Math. Soc. Jpn. 55, 335–364 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dorfmeister, J., Pedit, F., Wu, H.: Weierstrass type representation of harmonic maps into symmetric spaces. Commun. Anal. Geom. 6, 633–668 (1998)

    MathSciNet  MATH  Google Scholar 

  26. Dorfmeister, J., Wu, H.: Constant mean curvature surfaces and loop groups. J. Reine Angew. Math. 440, 1–47 (1993)

    MathSciNet  Google Scholar 

  27. Dorfmeister, J., Wu, H.: Construction of constant mean curvature trinoids from holomorphic potentials. Math. Z. 258, 773–803 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Douglas, J.: The mapping theorem of Koebe and the problem of Plateau. J. Math. Phys. 10, 106–130 (1930–1931)

    Google Scholar 

  29. Douglas, J.: Solution of the problem of Plateau. Trans. Am. Math. Soc. 33, 263–321 (1931)

    Article  Google Scholar 

  30. Douglas, J.: The higher topological form of the problem of Plateau. Ann. Sc. Norm. Super. Pisa 8, 195–218 (1939)

    Google Scholar 

  31. Douglas, J.: The most general form of the problem of Plateau. Am. J. Math. 61, 590–608 (1939)

    Article  Google Scholar 

  32. Duzaar, F., Steffen, K.: The Plateau problem for parametric surfaces with prescribed mean curvature. In: Jost, J. (ed.) Geometric Analysis and the Calculus of Variations, pp. 13–70. International Press, Cambridge (1996)

    Google Scholar 

  33. Finn, R.: On equations of minimal surface type. Ann. Math. 60, 397–416 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  34. Finn, R.: Remarks relevant to minimal surfaces and to surfaces of prescribed mean curvature. J. Anal. Math. 14, 139–160 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  35. Finn, R.: Equilibrium Capillary Surfaces. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  36. Fomenko, A., Tuzhilin, A.: Elements of the Geometry and Topology of Minimal Surfaces in Three-Dimensional Space. Translations of Mathematical Monographs, vol. 93. Am. Math. Soc., Providence (1991)

    MATH  Google Scholar 

  37. Fujimori, S., Kobayashi, S., Rossman, W.: In: Loop Groups Methods for Constant Mean Curvature Surfaces. Rokko Lectures in Mathematics, vol. 17 (2005). Kobe

    Google Scholar 

  38. de Gennes, P., Brochard-Wyart, F., Quéré, D.: Capillarity and Wetting Phenomena. Springer, New York (2004)

    Book  MATH  Google Scholar 

  39. Germain, S.: Mémoire sur la courbure des surfaces. J. Reine Angew. Math. 7, 1–29 (1831) (ed. A.L. Crelle)

    Article  MathSciNet  MATH  Google Scholar 

  40. Grosse-Brauckmann, K., Kusner, R.: Embedded constant mean curvature surfaces with special symmetry. Manuscr. Math. 99, 135–150 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Grosse-Brauckmann, K., Kusner, R., Sullivan, J.: Constant mean curvature surfaces with three ends. Proc. Natl. Acad. Sci. USA 97, 14067–14068 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  42. Grosse-Brauckmann, K., Kusner, R., Sullivan, J.: Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero. J. Reine Angew. Math. 564, 35–61 (2003)

    MathSciNet  MATH  Google Scholar 

  43. Grosse-Brauckmann, K., Kusner, R., Sullivan, J.: Coplanar constant mean curvature surfaces. Commun. Anal. Geom. 15, 985–1023 (2007)

    MathSciNet  MATH  Google Scholar 

  44. Gulliver, R., Spruck, J.: The Plateau problem for surfaces of prescribed mean curvature in a cylinder. Invent. Math. 13, 169–178 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gulliver, R., Spruck, J.: Existence theorems for parametric surfaces of prescribed mean curvature. Indiana Univ. Math. J. 22, 445–472 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  46. Heinz, E.: Über die Existenz einer Fläche konstanter mittlerer Krümmung belvorgegebener Berandung. Math. Ann. 127, 258–287 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  47. Heinz, E.: On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch. Ration. Mech. Anal. 35, 249–252 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hélein, F.: Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2001)

    Book  MATH  Google Scholar 

  49. Hilbert, D., Cohn-Vossen, S.: Geometry and the Imagination. AMS Chelsea Publ., Providence (1999)

    Google Scholar 

  50. Hildebrandt, S.: On the Plateau problem for surfaces of constant mean curvature. Commun. Pure Appl. Math. 23, 97–114 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  51. Hildebrandt, S., Kaul, H.: Two-dimensional variational problems with obstructions, and Plateau’s problem for H-surfaces in a Riemannian manifold. Commun. Pure Appl. Math. 25, 187–223 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  52. Hitchin, N.J.: Global differential geometry. In: Enquist, B., Schmid, W. (eds.) Mathematics Unlimited—2001 and Beyond, pp. 577–591. Springer, Heidelberg (2000)

    Google Scholar 

  53. Hoffman, D.: Global Theory of Minimal Surfaces. Clay Mathematics Proceedings, vol. 2. Am. Math. Soc., Providence (2005)

    MATH  Google Scholar 

  54. Hopf, H.: Differential Geometry in the Large. Lecture Notes in Mathematics, vol. 1000. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  55. Isenberg, C.: The Science of Soap Films and Soap Bubbles. Dover, New York (1992)

    Google Scholar 

  56. Jellet, J.H.: Sur la surface dont la courbure moyenne est constant. J. Math. Pures Appl. 18, 163–167 (1853)

    Google Scholar 

  57. Jleli, M.: End-to-end gluing of constant mean curvature hypersurfaces. Ann. Fac. Sci. Toulouse 18, 717–737 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. Jleli, M., Pacard, F.: An end-to-end construction for compact constant mean curvature surfaces. Pac. J. Math. 221, 81–108 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  59. Kapouleas, N.: Complete constant mean curvature surfaces in Euclidean three-space. Ann. Math. 131, 239–330 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  60. Kapouleas, N.: Compact constant mean curvature surfaces in Euclidean three-space. J. Differ. Geom. 33, 683–715 (1991)

    MathSciNet  MATH  Google Scholar 

  61. Kapouleas, N.: Constant mean curvature surfaces constructed by fusing Wente tori. Proc. Natl. Acad. Sci. USA 89, 5695–5698 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  62. Kapouleas, N.: Doubling and desingularization constructions for minimal surfaces. In: Surveys in Geometric Analysis and Relativity. Adv. Lect. Math., vol. 20, pp. 281–325. International Press, Somerville (2011)

    Google Scholar 

  63. Kapouleas, N., Yang, S.D.: Minimal surfaces in the three-sphere by doubling the Clifford torus. Am. J. Math. 132, 257–295 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  64. Karcher, H., Pinkall, U., Sterling, I.: New minimal surfaces in S 3. J. Differ. Geom. 28, 169–185 (1988)

    MathSciNet  MATH  Google Scholar 

  65. Kilian, M., Kobayashi, S.P., Rossman, W., Schmitt, N.: Constant mean curvature surfaces of any positive genus. J. Lond. Math. Soc. 72, 258–272 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  66. Kilian, M., McIntosh, I., Schmitt, N.: New constant mean curvature surfaces. Expo. Math. 9, 595–611 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  67. Kilian, M., Schmitt, N., Sterling, I.: Dressing CMC n-noids. Math. Z. 246, 501–519 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  68. Korevaar, N., Kusner, R.: The global structure of constant mean curvature surfaces. Invent. Math. 114, 311–332 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  69. Korevaar, N., Kusner, R., Ratzkin, J.: On the nondegeneracy of constant mean curvature surfaces. Geom. Funct. Anal. 16, 891–923 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  70. Korevaar, N., Kusner, R., Solomon, B.: The structure of complete embedded surfaces with constant mean curvature. J. Differ. Geom. 30, 465–503 (1989)

    MathSciNet  MATH  Google Scholar 

  71. Kusner, R., Mazzeo, R., Pollack, D.: The moduli space of complete embedded constant mean curvature surfaces. Geom. Funct. Anal. 6, 120–137 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  72. Langbein, D.: Capillary Surfaces: Shape—Stability—Dynamics in Particular Under Weightlessness. Springer, Berlin (2002)

    Book  Google Scholar 

  73. Laplace, P.S.: Traité de mécanique céleste; suppléments au Livre X. Oeuvres Complete, vol. 4. Gauthier-Villars, Paris (1805)

    Google Scholar 

  74. Lawson, H.B.: Complete minimal surfaces in S 3. Ann. Math. 92, 335–374 (1970)

    Article  MATH  Google Scholar 

  75. Lawson, H.B.: The unknottedness of minimal embeddings. Invent. Math. 11, 183–187 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  76. Lawson, H.B.: Lectures on Minimal Submanifolds, vol. I. Publish or Perish, Berkeley (1980)

    MATH  Google Scholar 

  77. Liebmann, H.: Über die Verbiebung der geschlossenen Flächen positiver Krümmung. Math. Ann. 53, 91–112 (1900)

    Article  MathSciNet  Google Scholar 

  78. Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. (2), to appear

    Google Scholar 

  79. Mazzeo, R., Pacard, F.: Constant mean curvature surfaces with Delaunay ends. Commun. Anal. Geom. 9, 169–237 (2001)

    MathSciNet  MATH  Google Scholar 

  80. Mazzeo, R., Pacard, F.: Bifurcating Nodoids. Topology and Geometry: Commemorating SISTAG. Contemp. Math., pp. 169–186. Am. Math. Soc., Providence (2002)

    Book  Google Scholar 

  81. Mazzeo, R., Pacard, F., Pollack, D.: Connected sums of constant mean curvature surfaces in Euclidean 3 space. J. Reine Angew. Math. 536, 115–165 (2001)

    MathSciNet  MATH  Google Scholar 

  82. Mazzeo, R., Pacard, F., Pollack, D.: The conformal theory of Alexandrov embedded constant mean curvature surfaces in \(\mathbb{R}^{3}\). In: Hofmann, D. (ed.) Global Theory of Minimal Surfaces. Clay Mathematics Proceedings, vol. 2, pp. 525–559. Am. Math. Soc., Providence (2005)

    Google Scholar 

  83. Meeks, W.H. III: The topology and geometry of embedded surfaces of constant mean curvature. J. Differ. Geom. 27, 539–552 (1988)

    MathSciNet  MATH  Google Scholar 

  84. Meeks, W.H. III, Yau, S.T.: The classical Plateau problem and the topology of three dimensional manifolds. Topology 21, 409–442 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  85. Meeks, W.H. III, Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z. 179, 151–168 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  86. Meusnier, M.: Mémoire sur la courbure des surfaces. Mémoires de Mathematique et de Physique (de savans étrangers) de l’Academie X, 447–550 (1785)

    Google Scholar 

  87. Montiel, S., Ros, A.: Minimal immersions of surfaces by the first eigenfunctions and conformal area. Invent. Math. 83, 153–166 (1985)

    Article  MathSciNet  Google Scholar 

  88. Montiel, S., Ros, A.: Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. In: Pitman Monogr. Surveys Pure Appl. Math., vol. 52, pp. 279–296. Longman Sci. Tech, Harlow (1991)

    Google Scholar 

  89. Nitsche, J.C.C.: Lectures on Minimal Surfaces, vol. 1. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  90. Norbury, P.: The 12th problem. Aust. Math. Soc. Gaz. 32, 244–246 (2005)

    Google Scholar 

  91. Osserman, R.: A Survey of Minimal Surfaces. Dover, New York (1986)

    Google Scholar 

  92. Pacard, F., Rosenberg, H.: Attaching handles to Delaunay nodoids (2010). arXiv:1010.4974

  93. Perdomo, O.: Embedded constant mean curvature hypersurfaces on spheres. Asian J. Math. 14, 73–108 (2010)

    MathSciNet  MATH  Google Scholar 

  94. Pinkall, U., Sterling, I.: On the classification of constant mean curvature tori. Ann. Math. 130, 407–451 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  95. Plateau, J.A.F.: Statique expérimentale et théoretique des liquides soumis aux seule forces moléculaires, vols. I, II. Gautier-Villars, Paris (1873)

    Google Scholar 

  96. Radó, T.: The problem of least area and the problem of Plateau. Math. Z. 32, 763–795 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  97. Radó, T.: On Plateau’s problem. Ann. Math. 31, 457–469 (1930)

    Article  MATH  Google Scholar 

  98. Radó, T.: Contributions to the theory of minimal surfaces. Acta Litt. Sci. Univ. Szeged 2, 1–20 (1932–1934)

    Google Scholar 

  99. Radó, T.: On the Problem of Plateau. Chelsea, New York (1951)

    Google Scholar 

  100. Ratzkin, J.: An end-to-end gluing construction for surfaces of constant mean curvature. Ph. Doctoral thesis, University of Washington (2001)

    Google Scholar 

  101. Reilly, R.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J. 26, 459–472 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  102. Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoam. 3, 447–453 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  103. Ros, A.: Compact hypersurfaces with constant scalar curvature and a congruence theorem. J. Differ. Geom. 27, 215–220 (1988)

    MathSciNet  MATH  Google Scholar 

  104. Ros, A.: A two-piece property for compact minimal surfaces in a three-sphere. Indiana Univ. Math. J. 44, 841–849 (1995)

    MathSciNet  MATH  Google Scholar 

  105. Schoen, R.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Commun. Pure Appl. Math. 41, 317–392 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  106. Schwarz, H.A.: Beweis des Satzes, dass die Kugel kleinere Oberfläche besitzt als jeder andere Körper gleichen Volumens. Nachrichten der Königlichen Gesellschaft für Wissenschaften Göttingen, pp. 1–13 (1884)

    Google Scholar 

  107. Serrin, J.: The problem of Dirichlet for quasilinear elliptic equations with many independent variables. Philos. Trans. R. Soc. Lond. Ser. A 264, 413–496 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  108. Spivak, M.: A Comprehensive Introduction to Differential Geometry. Publish Or Perish, Houston (1979)

    Google Scholar 

  109. Steffen, K.: On the existence of surfaces with prescribed mean curvature and boundary. Math. Z. 146, 113–135 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  110. Steffen, K.: On the nonuniqueness of surfaces with constant mean curvature spanning a given contour. Arch. Ration. Mech. Anal. 94, 101–122 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  111. Steffen, K.: Parametric surfaces of prescribed mean curvature. In: Hildebrandt, S., Struwe, M. (eds.) Calculus of Variations and Geometric Evolution Problems, Lectures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, June 15–22, 1996. Lecture Notes in Mathematics, vol. 1713, pp. 211–265. Springer, Berlin (1999)

    Chapter  Google Scholar 

  112. Steffen, K., Wente, H.C.: The non-existence of branch points in solutions to certain classes of Plateau type variational problems. Math. Z. 163, 211–238 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  113. Steiner, J.: Sur le maximum et le minimum des figures dans le plan, sur la sphére et dans l’espace en général. J. Reine Angew. Math. 24, 93–152 (1842)

    Article  MATH  Google Scholar 

  114. Struwe, M.: Large H-surfaces via the mountain-pass-lemma. Math. Ann. 270, 441–459 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  115. Struwe, M.: Nonuniqueness in the Plateau problem for surfaces of constant mean curvature. Arch. Ration. Mech. Anal. 93, 135–157 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  116. Struwe, M.: Plateau’s Problem and the Calculus of Variations. Mathematical Notes. Princeton Univ. Press, Princeton (1988)

    Google Scholar 

  117. Struwe, M.: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Berlin (2000)

    Google Scholar 

  118. Umehara, M., Yamada, K.: A deformation of tori with constant mean curvature in \(\mathbb{R}^{3}\) to those in other space forms. Trans. Am. Math. Soc. 330, 845–857 (1992)

    MathSciNet  MATH  Google Scholar 

  119. Urbano, F.: Minimal surfaces with low index in the three-dimensional sphere. Proc. Am. Math. Soc. 108, 989–992 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  120. Walter, R.: Constant mean curvature tori with spherical curvature lines in non-Euclidean geometry. Manuscr. Math. 63, 343–363 (1989)

    Article  MATH  Google Scholar 

  121. Wente, H.C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318–344 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  122. Wente, H.C.: A general existence theorem for surfaces of constant mean curvature. Math. Z. 120, 277–288 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  123. Wente, H.C.: Large solutions to the volume constrained Plateau problem. Arch. Ration. Mech. Anal. 75, 59–77 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  124. Wente, H.C.: Counterexample to a conjecture of H. Hopf. Pac. J. Math. 121, 193–243 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  125. Werner, H.: Problem von Douglas für Flächen konstanter mittlerer Krümmung. Math. Ann. 135, 303–319 (1957)

    Article  Google Scholar 

  126. Young, T.: An essay of the cohesion of fluids. Philos. Trans. R. Soc. London 95, 65–87 (1805)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Geometry and Topology, University of Granada, Granada, Spain

    Rafael López

Authors
  1. Rafael López
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

López, R. (2013). A Brief Historical Introduction and Motivations. In: Constant Mean Curvature Surfaces with Boundary. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39626-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation