Principal lines on surfaces immersed with constant mean curvature
HTML articles powered by AMS MathViewer
- by C. Gutiérrez and J. Sotomayor PDF
- Trans. Amer. Math. Soc. 293 (1986), 751-766 Request permission
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
- V. Arnol′d, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, “Mir”, Moscow, 1980 (French). Translated from the Russian by Djilali Embarek. MR 626685
- Shiing Shen Chern, On surfaces of constant mean curvature in a three-dimensional space of constant curvature, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 104–108. MR 730266, DOI 10.1007/BFb0061413
- Carlos Gutiérrez, Structural stability for flows on the torus with a cross-cap, Trans. Amer. Math. Soc. 241 (1978), 311–320. MR 492303, DOI 10.1090/S0002-9947-1978-0492303-2
- J. Sotomayor and C. Gutierrez, Structurally stable configurations of lines of principal curvature, Bifurcation, ergodic theory and applications (Dijon, 1981) Astérisque, vol. 98, Soc. Math. France, Paris, 1982, pp. 195–215. MR 724448
- C. Gutiérrez and J. Sotomayor, An approximation theorem for immersions with stable configurations of lines of principal curvature, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 332–368. MR 730276, DOI 10.1007/BFb0061423
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
- John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221–274. MR 523212, DOI 10.1007/BF02395062
- David A. Hoffman, Surfaces of constant mean curvature in manifolds of constant curvature, J. Differential Geometry 8 (1973), 161–176. MR 390973
- David A. Hoffman and Robert Osserman, The Gauss map of surfaces in $\textbf {R}^{n}$, J. Differential Geom. 18 (1983), no. 4, 733–754 (1984). MR 730925
- Heinz Hopf, Differential geometry in the large, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1983. Notes taken by Peter Lax and John Gray; With a preface by S. S. Chern. MR 707850, DOI 10.1007/978-3-662-21563-0 —, Über Flächen mit einer Relation zwischen den Hauptkrümungen, Math. Nachr. 4B ( 1950/51 ).
- Wu-yi Hsiang, Zhen Huan Teng, and Wen Ci Yu, New examples of constant mean curvature immersions of $(2k-1)$-spheres into Euclidean $2k$-space, Ann. of Math. (2) 117 (1983), no. 3, 609–625. MR 701257, DOI 10.2307/2007036
- James A. Jenkins, On the local structure of the trajectories of a quadratic differential, Proc. Amer. Math. Soc. 5 (1954), 357–362. MR 62227, DOI 10.1090/S0002-9939-1954-0062227-2
- James A. Jenkins, A general coefficient theorem, Trans. Amer. Math. Soc. 77 (1954), 262–280. MR 64146, DOI 10.1090/S0002-9947-1954-0064146-9
- Katsuei Kenmotsu, The Weierstrass formula for surfaces of prescribed mean curvature, Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977) North-Holland, Amsterdam-New York, 1979, pp. 73–76. MR 574254
- Robert Osserman, A survey of minimal surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne, 1969. MR 0256278
- M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120. MR 142859, DOI 10.1016/0040-9383(65)90018-2 E. Picard, Traité d’analyse, Vol. 3, Gauthier-Villars, Paris, 1908. M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Berkeley, Calif., 1979. K. Strebel, On quadratic differentials and extremal quasiconformal mappings, Lecture Notes, Univ. of Minnesota, 1967. G. Valiron, Équations fonctionelles applications, 2$^{e}$ ed., Masson, Paris, 1950 (Chapitre III).
- Henry C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), no. 1, 193–243. MR 815044, DOI 10.2140/pjm.1986.121.193
- Joseph A. Wolf, Surfaces of constant mean curvature, Proc. Amer. Math. Soc. 17 (1966), 1103–1111. MR 200879, DOI 10.1090/S0002-9939-1966-0200879-2
- Shing Tung Yau (ed.), Seminar on Differential Geometry, Annals of Mathematics Studies, No. 102, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979–1980. MR 645728
Additional Information
- © Copyright 1986 American Mathematical Society
- 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