Pacific Journal of Mathematics

Comparison surfaces for the Willmore problem.

Rob Kusner

Article information

Pacific J. Math., Volume 138, Number 2 (1989), 317-345.

First available in Project Euclid: 8 December 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 53A30: Conformal differential geometry 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 58E12: Applications to minimal surfaces (problems in two independent variables) [See also 49Q05]


Kusner, Rob. Comparison surfaces for the Willmore problem. Pacific J. Math. 138 (1989), no. 2, 317--345.

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  • [BT] T. Banchoff, Triple points and surgery of immersed surfaces, Proc. Amer. Math. Soc, 46 (1974), 407-413.
  • [BR1] R. Bryant, A duality theoremfor Willmore surfaces, J. Differential Geom., 20 (1984), 23-53.
  • [BR2] R. Bryant, Personal communication.
  • [CBY] B.-Y. Chen, Geometry of Submanifolds, Marcel Dekker, New York, 1973.
  • [FG] G. Francis, Some equivariant eversions of the sphere, Privately circulated manuscript, 1977.
  • [GT] D. Gilbarg and N. Trudinger, Elliptic PartialDifferentialEquations of Second Order (2nd edition), Springer-Verlag, Berlin, 1983.
  • [GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Inter- science, New York, 1978.
  • [GOR] R. Gulliver, R. Osserman and H. Royden, A theory of branchedimmersions of surfaces,Amer. J. Math., 95 (1973), 750-812.
  • [HH] J. Hass and J. Hughes, Immersions of surfaces in 3-manifolds,Topology, 24 (1985), 97-112.
  • [HJ] J. Hughes, Immersion Theory, Dissertation, Univ. of California, Berkeley.
  • [KWP] W. Kuhnel and U. Pinkall, On total mean curvatures, Quart. J. Math. Oxford, 37 (1986), 437-448.
  • [KRl] R. Kusner, Conformal geometry and complete minimal surfaces,Bull. Amer. Math. Soc, 17 (1987), 291-295.
  • [KR2] R. Kusner, Global geometry of extremal surfaces in three-space, Dissertation, Univ. of California, Berkeley.
  • [KR3] R. Kusner, The Willmore problem in R3 and R4, to appear.
  • [LHB] H. B. Lawson, Jr., Complete minimal surfaces in S4, Ann. of Math., 92 (1970), 335-374.
  • [LY] P. Li and S.-T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math., 69(1982), 269-291.
  • [MW1] W. Meeks III,The classification of complete minimal surfacesin R3 with total curvaturegreater than -8, Duke Math. J., 48 (1981), 523-535.
  • [MW2] W. Meeks III, The topological uniqueness of minimal surfaces in Euclidean three space,Topology, 20 (1981), 389-410.
  • [PU] U. Pinkall, Regular homotopy classes of immersed surfaces, Topology, 24 (1985), 421-434.
  • [PS] U. Pinkall and I. Sterling, Willmore surfaces,Math. Intellegencer (1987).
  • [SL] L. Simon, Existence of Willmore surfaces,Proc. Aust. Nat. Univ., 1986.
  • [SM] M. Spivak, A comprehensiveintroduction to differential geometry IV, Publish or Perish, Berkeley, 1975.
  • [TG] G.Thomsen, Uber Konforme geometrie I: Grundlagender konformen flachen- theorie,Abh. Math. Sem. Hamburg, (1923), 31-56.
  • [WJ] J. Weiner, On a problem of Chen, Willmore, et ah, Indiana Univ. Math. J., 27(1978), 19-35.
  • [WR] R. Wells, Cobordismgroups of immersions, Topology, 5 (1966), 281-294.
  • [WT] T. Willmore, Note on embedded surfaces,Anal. Stunt, ale Univ. Iasi Sect. I. a Math., 11 (1965), 493-496.