Immersed surfaces of prescribed Gauss curvature into Minkowski space

Ge, Yuxin

doi:10.1090/S0002-9939-00-05770-1

Immersed surfaces of prescribed Gauss curvature into Minkowski space
HTML articles powered by AMS MathViewer

by Yuxin Ge PDF

Proc. Amer. Math. Soc. 129 (2001), 2093-2101 Request permission

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

S. I. Al′ber, Spaces of mappings into a manifold of negative curvature, Dokl. Akad. Nauk SSSR 178 (1968), 13–16 (Russian). MR 0230254
J. Eells and L. Lemaire, A report on the harmonic maps, Bull. London. Math. Soc 10 (1978), 1-68.
Y. Ge, An elliptic variational approach to immersed surfaces of prescribed Gauss curvature, Calc. Var. 7 (1998) 173-190.
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
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, DOI 10.1007/978-3-642-61798-0
Philip Hartman, On homotopic harmonic maps, Canadian J. Math. 19 (1967), 673–687. MR 214004, DOI 10.4153/CJM-1967-062-6
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
F. Hélein, Applications harmoniques, lois de conservation et repère mobile, Diderot éditeur, Paris-New York-Amsterdam (1996).
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
J. Jost and M. Meier, Boundary regularity for minima of certain quadratic functionals, Math. Ann. 262 (1983), no. 4, 549–561. MR 696525, DOI 10.1007/BF01456068
H. Lewy, On differential geometry in the large, I (Minkowski’s problem), Trans. Amer. Math. Soc. 43 (1938) 258-270.
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
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Stoïlow, Leçons sur les principes topologiques de la théorie des fonctions analytiques, Paris (1938), Gauthier-Villars, p. 130.
M. Struwe, Variational Methods, Springer, Berlin-Heidelberg-New York-Tokyo (1990).