Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Principal lines on surfaces immersed with constant mean curvature


Authors: C. Gutiérrez and J. Sotomayor
Journal: Trans. Amer. Math. Soc. 293 (1986), 751-766
MSC: Primary 53A10; Secondary 58F18
DOI: https://doi.org/10.1090/S0002-9947-1986-0816323-5
MathSciNet review: 816323
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Configurations of lines of principal curvature on constant mean curvature immersed surfaces are studied. Analytical models for these configurations near general isolated umbilical points and particular types of ends are found. From the existence of transversal invariant measures for the foliations by principal lines, established here, follows that the union of recurrent lines of principal curvature is an open set. Examples illustrating all possible cases are given.


References [Enhancements On Off] (What's this?)

  • [1] V. Arnold, Chapitres supplementaires de la théorie des équations différentielles ordinaires, "Mir," Moscow, 1980. MR 626685 (83a:34003)
  • [2] S. S. Chern, On surfaces of constant mean curvature in a three-dimensional space of constant curvature, Geometric Dynamics (Proc. Rio de Janeiro, 1981), Lecture Notes in Math., vol. 1007, Springer-Verlag, 1983, pp. 104-108. MR 730266 (86b:53058)
  • [3] C. Gutierrez, Structural stability for flows on the torus with a cross-cap, Trans. Amer. Math. Soc. 241 (1978), 311-320. MR 492303 (80k:58065)
  • [4] C. Gutierrez and J. Sotomayor, Structurally stable configurations of lines of principal curvature, Astérisque 98-99 (1982), 195-215. MR 724448 (85h:58006)
  • [5] -, An approximation theorem for immersions with stable configurations of lines of principal curvature, Geometric Dynamics (Proc. Rio de Janeiro, 1981), Lectures Notes in Math., vol. 1007, Springer-Verlag, 1983, pp. 332-368. MR 730276 (85b:53002)
  • [6] P. Hartman, Ordinary differential equations, Wiley, New York, 1964. MR 0171038 (30:1270)
  • [7] J. Hubbard and H. Masur, Quadratic differential and foliations, Acta Math. 142 (1979), 221-273. MR 523212 (80h:30047)
  • [8] D. A. Hoffman, Surfaces of constant mean curvature in manifolds of constant curvature, J. Differential Geom. 8 (1973), 161-176. MR 0390973 (52:11796)
  • [9] D. A. Hoffman and R. Osserman, The Gauss map of surfaces in $ {R^n}$, J. Differential Geom. 18 (1983), 733-754. MR 730925 (85i:53059)
  • [10] H. Hopf, Lectures on differential geometry in the large, Notes by J. M. Gray, Stanford Univ., 1954. Reprinted in Lecture Notes in Math., vol. 1000, Springer-Verlag. MR 707850 (85b:53001)
  • [11] -, Über Flächen mit einer Relation zwischen den Hauptkrümungen, Math. Nachr. 4B ( 1950/51 ).
  • [12] W.-Y. Hsiang, Z.-H. Teng and W.-C. Yu, New examples of constant mean curvature immersions of $ 3$-sphere into Euclidean $ 4$-space, Ann. of Math. (2) 117 (1983), 609-625. MR 701257 (84i:53057)
  • [13] J. A. Jenkins, On the local structure of the trajectories of a quadratic differential, Proc. Amer. Math. Soc. 5 (1954), 357-362. MR 0062227 (15:947d)
  • [14] -, A general coefficient theorem, Trans. Amer. Math. Soc. 77 (1954), 262-280. MR 0064146 (16:232f)
  • [15] K. Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979), 89-99. MR 552581 (81c:53005b)
  • [16] R. Osserman, A survey of minimal surfaces, Van Nostrand Reinhold, New York, 1969. MR 0256278 (41:934)
  • [17] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101-120. MR 0142859 (26:426)
  • [18] E. Picard, Traité d'analyse, Vol. 3, Gauthier-Villars, Paris, 1908.
  • [19] M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Berkeley, Calif., 1979.
  • [20] K. Strebel, On quadratic differentials and extremal quasiconformal mappings, Lecture Notes, Univ. of Minnesota, 1967.
  • [21] G. Valiron, Équations fonctionelles applications, 2$ ^{e}$ ed., Masson, Paris, 1950 (Chapitre III).
  • [22] H. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. (to appear). MR 815044 (87d:53013)
  • [23] J. A. Wolf, Surfaces of constant mean curvature. Proc. Amer. Math. Soc. 17 (1966), 1103-1111. MR 0200879 (34:765)
  • [24] S. T. Yau, Seminar on Differential Geometry, Ann. of Math. Studies, no. 102, Princeton Univ. Press, Princeton, N. J., 1982. MR 645728 (83a:53002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53A10, 58F18

Retrieve articles in all journals with MSC: 53A10, 58F18


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0816323-5
Keywords: Principal lines, constant mean curvature surfaces, umbilical points, ends, quadratic differentials, transversal measures
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society