Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Intrinsic curvature of the induced metric on harmonically immersed surfaces


Author: Tilla Klotz Milnor
Journal: Proc. Amer. Math. Soc. 94 (1985), 549-552
MSC: Primary 53C50; Secondary 53C42, 58E20
DOI: https://doi.org/10.1090/S0002-9939-1985-0787911-4
MathSciNet review: 787911
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A theorem by Wissler is used to prove the following result. Suppose that an oriented surface $ S$ with indefinite prescribed metric $ h$ is harmonically mapped into an arbitrary pseudo-Riemannian manifold so that the metric $ I$ induced on $ S$ is complete and Riemannian. Then the intrinsic curvature $ K\left( I \right)$ of the immersion satisfies $ {\text{inf}}\left\vert {K\left( I \right)} \right\vert = 0$, with $ {\text{sup}}\left\vert {{\text{grad 1/K}}\left( I \right) = \infty } \right.$ in case $ K\left( I \right)$ never vanishes on $ S$.


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

  • [1] R. L. Bishop and R. J. Crittenden, Geometry of manifolds, Academic Press, New York, 1964. MR 0169148 (29:6401)
  • [2] N. V. Efimov, Appearance of singularities on surfaces of negative curvature, Mat. Sb. 64 (1964), 286-320; English Transl., Amer. Math. Soc. Transl. 66 (1968), 154-190. MR 0167938 (29:5203)
  • [3] -, Differential criteria for homeomorphism of certain mappings with applications to the theory of surfaces, Mat. Sb. 76 (1968), 499-512; English Transl., Math USSR-Sb. 5 (1968), 475-488. MR 0230258 (37:5821)
  • [4] -, Surfaces with slowly changing negative curvature, Uspekhi Mat. Nauk 21 (1966), 3-58; English Transl. Russian Math. Surveys 21 (1966), 1-56. MR 0202092 (34:1966)
  • [5] C. H. Gu, On the Cauchy problem for harmonic maps defined on two dimensional Minkowski space, Comm. Pure Appl. Math. 33 (1980), 727-738. MR 596432 (82g:58027)
  • [6] D. Hilbert, Grundlagen der Geometrie, vol. 9, Stuttgart, 1962. MR 0177322 (31:1585)
  • [7] E. Holmgren, Sur les surfaces a courbre constante negative, C.R. Acad. Sci. Paris 13 4 (1902), 740-743.
  • [8] T. K. Milnor, Efimov's theorem about complete immersed surfaces of negative curvature, Adv. in Math. 8 (1972), 474-543. MR 0301679 (46:835)
  • [9] -, Abstract Weingarten surfaces, J. Differential Geometry 15 (1980), 365-380. MR 620893 (82i:53048)
  • [10] -, Harmonic maps and classical surface theory in Minkowski $ 3$-space, Trans. Amer. Math. Soc. 280 (1983), 161-185. MR 712254 (85e:58037)
  • [11] B. O'Neill, Semi-Riemannian geometry, Academic Press, New York, 1983.
  • [12] K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), 207-221. MR 0408535 (53:12299)
  • [13] J. J. Stoker, Differential geometry, Wiley-Interscience, New York, 1969. MR 0240727 (39:2072)
  • [14] C. Wissler, Globale Tschebyscheff-Netze auf Riemannischen Mannigfaltkeiten und Fortsetzung von Flächen konstanter negativer Krummung, Comm. Math. Helv. 47 (1972), 348-372. MR 0320968 (47:9501)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C50, 53C42, 58E20

Retrieve articles in all journals with MSC: 53C50, 53C42, 58E20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0787911-4
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society