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Maximal spacelike surfaces in a certain homogeneous Lorentzian 3-manifold


The 2-parameter family of certain homogeneous Lorentzian 3-manifolds, which includes Minkowski 3-space and anti-de Sitter 3-space, is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula for maximal spacelike surfaces in the homogeneous Lorentzian 3-manifolds is obtained. The normal Gauß map of maximal spacelike surfaces and its harmonicity are discussed.

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  1. 1.

    From here on we mean a surface by an immersion.

  2. 2.

    It can be also obtained directly from (14).


  1. 1.

    Bondi, H., Gold, T.: The steady-state theory of the expanding universe. Mon. Not. Roy. Ast. Soc. 108, 252–270 (1948)

    Article  Google Scholar 

  2. 2.

    de Lira, J.H.S., Hinojosa, J.A.: The Gauss map of minimal surfaces in the Anti-de Sitter space. J. Geom. Phys. 61, 610–623 (2011)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Eells, J., Lemaire, L.: Selected topics in harmonic maps, C.M.S. Regional Conference Series 50. Amer. Math. Soc. (1983)

  4. 4.

    Góes, C.C., Simões, P.A.Q.: The generalized Gauss map of minimal surfaces in \(H^3\) and \(H^4\). Bol. Soc. Brasil Mat. 18, 35–47 (1987)

  5. 5.

    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge Univ. Press, Cambridge (1973)

    Book  Google Scholar 

  6. 6.

    Hoyle, F.: A new model for the expanding universe. Mon. Not. Roy. Ast. Soc. 108, 372–382 (1948)

    Article  Google Scholar 

  7. 7.

    Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups. Chin. Ann. Math. B. 24, 73–84 (2003)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups II. Bull. Aust. Math. Soc. 73, 365–374 (2006)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Inoguchi, J., Lee, S.: A Weierstrass type representation for minimal surfaces in Sol. Proc. Am. Math. Soc. 136, 2209–2216 (2008)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kobayashi, O.: Maximal surfaces in the 3-dimensional Minkowski space. Tokyo J. Math. 6, 297–309 (1983)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Kokubu, M.: Weierstrass representation for minimal surfaces in hyperbolic space. Tôhoku Math. J. 49, 367–377 (1997)

  12. 12.

    Lee, S.: Maximal surfaces in a certain 3-dimensional homogeneous spacetime. Differ. Geom. Appl. 26(5), 536–543 (2008)

    MathSciNet  Article  Google Scholar 

  13. 13.

    McNertney, L.: One-parameter families of surfaces with constant curvature in Lorentz 3-space, Ph. D. Thesis, Brown Univ., Providence, RI, USA (1980)

  14. 14.

    Mercuri, F., Montaldo, S., Piu, P.: A Weierstrass representation formula for minimal surfaces in \({\mathbb{H}}_3\) and \({\mathbb{H}}^2\times {\mathbb{R}}\). Acta Math. Sin. (Engl. Ser.), 22 (2006)

  15. 15.

    Nelli, B., Rosenberg, H.: Minimal surfaces in \({\mathbb{H}}^2\times {\mathbb{R}}\). Bull. Brasil Math. Soc. (N.S.) 33, 263–292 (2002)

  16. 16.

    Rosenberg, H.: Minimal surfaces in \(M\times R\). Illinois J. Math. 46(4), 1177–1195 (2002)

  17. 17.

    Thurston, W.M.: Three-dimensional Geometry and Topology I. In: Levy, S. (ed.) Princeton Math. Series, vol. 35 (1997).

  18. 18.

    Uhlenbeck, K.: Harmonic maps into Lie groups (classical solutions of the chiral model). J. Differ. Geom. 30, 1–50 (1989)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Wood, J.C.: Harmonic maps into symmetric spaces and integrable systems. In: Aspects of Mathematics, vol. E23, pp. 29–55. Vieweg, Braunschweig/Wiesbaden (1994)

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Correspondence to Sungwook Lee.

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Lee, S. Maximal spacelike surfaces in a certain homogeneous Lorentzian 3-manifold. J. Geom. 112, 27 (2021).

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  • Anti-de Sitter space
  • harmonic map
  • homogeneous manifold
  • Lorentzian manifold
  • maximal surface
  • Minkowski space
  • spacelike surface
  • solvable Lie group

Mathematics Subject Classification

  • 53A10
  • 53C30
  • 53C42
  • 53C50