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)



Immersed surfaces of prescribed Gauss curvature into Minkowski space

Author: Yuxin Ge
Journal: Proc. Amer. Math. Soc. 129 (2001), 2093-2101
MSC (2000): Primary 53C42, 53B25
Published electronically: December 7, 2000
MathSciNet review: 1825922
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a positive real valued function $k(x)$ on the disc, we will immerse the disc into three dimensional Minkowski space in such a way that Gauss curvature at the image point of $x$ is $-k(x)$. Our approach lies on the construction of Gauss map of surfaces.

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

  • 1. S. I. Al′ber, Spaces of mappings into a manifold of negative curvature, Dokl. Akad. Nauk SSSR 178 (1968), 13–16 (Russian). MR 0230254
  • 2. J. Eells and L. Lemaire, A report on the harmonic maps, Bull. London. Math. Soc 10 (1978), 1-68.
  • 3. Y. Ge, An elliptic variational approach to immersed surfaces of prescribed Gauss curvature, Calc. Var. 7 (1998) 173-190.
  • 4. Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
  • 5. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • 6. Philip Hartman, On homotopic harmonic maps, Canad. J. Math. 19 (1967), 673–687. MR 0214004
  • 7. Philip Hartman and Aurel Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math. 75 (1953), 449–476. MR 0058082
  • 8. F. Hélein, Applications harmoniques, lois de conservation et repère mobile, Diderot éditeur, Paris-New York-Amsterdam (1996).
  • 9. Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. MR 1100926
  • 10. J. Jost and M. Meier, Boundary regularity for minima of certain quadratic functionals, Math. Ann. 262 (1983), no. 4, 549–561. MR 696525, 10.1007/BF01456068
  • 11. H. Lewy, On differential geometry in the large, I (Minkowski's problem), Trans. Amer. Math. Soc. 43 (1938) 258-270. CMP 95:18
  • 12. Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
  • 13. Louis Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394. MR 0058265
  • 14. Stoïlow, Leçons sur les principes topologiques de la théorie des fonctions analytiques, Paris (1938), Gauthier-Villars, p. 130.
  • 15. M. Struwe, Variational Methods, Springer, Berlin-Heidelberg-New York-Tokyo (1990).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C42, 53B25

Retrieve articles in all journals with MSC (2000): 53C42, 53B25

Additional Information

Yuxin Ge
Affiliation: Département de Mathématiques, Faculté de Sciences et Technologie, Université Paris XII-Val de Marne, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France; C.M.L.A., E.N.S de Cachan, 61, avenue du Président Wilson, 94235 Cachan Cedex, France

Keywords: Gauss curvature, surfaces, Minkowski space, harmonic maps
Received by editor(s): April 29, 1999
Received by editor(s) in revised form: October 20, 1999
Published electronically: December 7, 2000
Communicated by: Bennett Chow
Article copyright: © Copyright 2000 American Mathematical Society