## Tohoku Mathematical Journal

### Minimal maps between the hyperbolic discs and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space

#### Abstract

Problems related to minimal maps are studied. In particular, we prove an existence result for the Dirichlet problem at infinity for minimal diffeomorphisms between the hyperbolic discs. We also give a representation formula for a minimal diffeomorphism between the hyperbolic discs by means of the generalized Gauss map of a complete maximal surface in the anti-de Sitter 3-space.

#### Article information

Source
Tohoku Math. J. (2), Volume 52, Number 3 (2000), 415-429.

Dates
First available in Project Euclid: 3 May 2007

https://projecteuclid.org/euclid.tmj/1178207821

Digital Object Identifier
doi:10.2748/tmj/1178207821

Mathematical Reviews number (MathSciNet)
MR2002e:58025

Zentralblatt MATH identifier
0978.53106

#### Citation

Aiyama, Reiko; Akutagawa, Kazuo; Wan, Tom Y. H. Minimal maps between the hyperbolic discs and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space. Tohoku Math. J. (2) 52 (2000), no. 3, 415--429. doi:10.2748/tmj/1178207821. https://projecteuclid.org/euclid.tmj/1178207821

#### References

• [1] R AIYAMA AND K. AKUTAGAWA, Kenmotsu-Bryant type representation formula for constant mean curva-ture spacelike surfaces \nH(-c2}, Differential Geom Appl. 9 (1998), 251-272
• [2] R AIYAMA AND K AKUTAGAWA, Kenmotsu type representation formula for surfaces with prescribed mea curvature in the 3-sphere, Tohoku Math J 52 (2000), 95-105.
• [3] K AKUTAGAWA, Harmonic diffeomorphisms of the hyperbolic plane, Trans Amer Math Soc 342 (1994), 325-342
• [4] K AKUTAGAWA AND S NISHIKAWA, The Gauss map and spacelike surfaces with prescribed mean curvatur in Minkowski 3-space, Tohoku Math J 42 (1990), 67-82
• [5] H I CHOI AND A TREIBERGS, Gauss mapof spacelike constant mean curvature hypersurfaces of Minkowsk space, J Differential Geom. 32 (1990), 775-817.
• [6] J EELLS AND L LEMAIRE, Selected topics in harmonic maps, C B. M. S Regional Conf Series, no 50, Amer Math Soc., Providence, R. I 1983
• [7] T. ISHIHARA, The harmonic Gauss maps in a generalized sense, J London Math Soc 26 (1982), 104-11
• [8] K. KENMOTSU, Weierstrass formula for surfaces of prescribed mean curvature, Math Ann. 245 (1979), 89 99
• [9] Y I LEE, Lagrangian minimal surfaces in Kahler-Einstein surfaces of negative scalar curvature, Comm Ana Geom 2 (1994), 579-592
• [10] P Li AND L -F. TAM, The heat equation and harmonic maps of complete manifolds, Invent Math 105 (1991), 1-46
• [11] P Li AND L -F TAM, Uniqueness and regularity of proper harmonic maps, Ann of Math 137 (1993), 167 201
• [12] P Li AND L -F TAM, Uniqueness and regularity of proper harmonic maps II, Indiana Univ Math J 4 (1993), 591-635
• [13] T K MILNOR, Harmonic maps and classical surface theory in Minkowski 3-space, Trans Amer Math So 280(1983), 161-185
• [14] B PALMER, Surfaces in Lorentzian hyperbolic space, Ann. Global Anal Geom 9 (1991), 117-12
• [15] E A. RUH AND J VILMS, The tension field of the Gauss map, Trans Amer Math Soc 149 (1970), 569-57
• [16] R SCHOEN, The role of harmonic mappings in rigidity and deformation problems, Complex Geometr (Osaka, 1990), 179-200, Lecture Notes in Pure and Appl Math, 143, Dekker, New York, 1993
• [17] L -F TAM AND T Y H WAN, Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribe Hopf differentials, Comm Anal Geom 2 (1994), 593-625
• [18] L -F TAM AND T Y H WAN, Quasi-conformal harmonic diffeomorphisms and the universal Teichmlle space, ! Differential Geom 42 (1995), 368-410
• [19] T Y H WAN, Constant mean curvature surface, harmonic maps and universal Teichmuller space, J Differ ential Geom 35 (1992), 643-657
• [20] T Y H WAN AND T K K Au, Parabolic constant mean curvature spacelike surfaces, Proc Amer Mat Soc 120(1994), 559-564