The Journal of Geometric Analysis

, Volume 8, Issue 4, pp 655–671 | Cite as

Real solvability of the equation ∂ 2/z ω = ρg and the topology of isolated umbilics

  • Brian Smyth
  • Frederico Xavier
Article

Abstract

The geometric form of a conjecture associated with the names of Loewner and Carathéodory states that near an isolated umbilic in a smooth surface in ℝ3, the principal line fields must have index ≤ 1. Real solutions of the differential equation ∂ 2/z ω = g, where the complex function g is given only up to multiplication by a positive function, are intimately related to umbilics. We determine necessary and sufficient conditions of an integral nature for real solvability of this equation, which is really a system of two wave equations. We then construct germs of line fields of every index j ∈ 1/2 ℤ on S2 that cannot be realized as the Gauss image of the principal line fields near an isolated umbilic of positive curvature on any smooth surface in ℝ3. These include the standard dipole line field of index two and controlled distortions of it.

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References

  1. [1]
    Akhiezer, N.I.The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965.Google Scholar
  2. [2]
    Andronov, A.A., Leontovich, E.A., Gordan, I.I., and Maier, A.G.Qualitative Theory of Second Order Dynamical Systems, John Wiley, New York. 1973.Google Scholar
  3. [3]
    Arnol’d, V.I. and Il’yashenko, Y.S.Ordinary Differential Equations, Vol. 1, Dynamical Systems, Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin. 1988.Google Scholar
  4. [4]
    Bol, G. Über Nabelpunkte auf einer Eifläche,Math. Z.,49, 389–410, (1943-44).CrossRefMathSciNetGoogle Scholar
  5. [5]
    Bonnet, O. Mémoire sur l’emploi d’un nouveau système de variables dans l’étude des propriétés des surfaces courbes,J. de Liouville,5, 153–266, (1860).Google Scholar
  6. [6]
    Darboux, G.Lecons sur la Théorie Générale des Surfaces, Vol. I, Gauthier-Villars, Paris, 1914.MATHGoogle Scholar
  7. [7]
    Evans, L.C. and Gariepy, R.R.Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1990.Google Scholar
  8. [8]
    Federer, H.Geometric Measure Theory, Springer-Verlag, Berlin, 1969.MATHGoogle Scholar
  9. [9]
    Hamburger, H. Beweis einer Carathéodorischen Vermütung, I,Ann. Math.,41, 63–86, (1940).CrossRefMathSciNetGoogle Scholar
  10. [10]
    Hamburger, H. Beweis einer Carathéodorischen Vermütung, II and III,Acta Math.,73, 174–332, (1941).CrossRefGoogle Scholar
  11. [11]
    Klotz, T. On G.Bol’s proof of the Carathéodory Conjecture,Comm. Pure Appl. Math.,12, 277–311, (1959).CrossRefMathSciNetMATHGoogle Scholar
  12. [12]
    Koosis, P. The logarithmic integral, I,Cambridge Studies in Advanced Mathematics,12, Cambridge University Press, Cambridge, (1988).Google Scholar
  13. [13]
    Lang, M. Nabelpunkte, Krümmungslinien, Brennflächen und ihre Metamorphosen, Dissertation, Technische Hochschule Darmstadt, Darmstadt, (1990).Google Scholar
  14. [14]
    Lefschetz, S.Differential Equations: Geometric Theory, Interscience Publishers, New York, 1963.MATHGoogle Scholar
  15. [15]
    Loewner, Ch. Conservation laws in compressible fluid flow and associated mappings,J. Rat. Mech. Anal.,2, 537–561, (1953).MathSciNetGoogle Scholar
  16. [16]
    Protter, H.M. and Weinberger, H.F.Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967.Google Scholar
  17. [17]
    Smyth, B. and Xavier, F. A. sharp geometric estimate for the index of an umbilic on a smooth surface,Bull. London Math. Soc.,24, 176–180, (1992).CrossRefMathSciNetMATHGoogle Scholar
  18. [18]
    Tenenblatt, K. and Terng, C.L. Bäcklund’s theorem for n-dimensional submanifolds of ℝ2n−1,Ann. Math.,111, 477–490, (1980).CrossRefGoogle Scholar
  19. [19]
    Titus, C.J. A proof of a conjecture of Loewner and of the conjecture of Carathéodory on umbilic points,Acta Math.,131, 43–77, (1973).CrossRefMathSciNetMATHGoogle Scholar
  20. [20]
    Yau, S.T.Seminar on Differential Geometry, Princeton University Press, Princeton, NJ, 1982.MATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1998

Authors and Affiliations

  • Brian Smyth
    • 1
  • Frederico Xavier
    • 1
  1. 1.Mathematics DepartmentUniversity of Notre DameNotre Dame

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