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A minimal immersion of the hyperbolic plane into the neutral pseudo-hyperbolic 4-space and its characterization

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Abstract

It is well known that there are no minimal surfaces of constant curvature lying fully in the hyperbolic 4-space H 4(−1). In contrast, in this article we discover a minimal immersion of the hyperbolic plane \({H^2(-\frac{1}{3})}\) of curvature \({-\frac{1}{3}}\) into the neutral pseudo-hyperbolic 4-space \({H^4_2(-1)}\). Moreover, we prove that, up to rigid motions of \({H^4_2(-1)}\), this minimal immersion provides the only space-like parallel minimal surface lying fully in \({H^4_2(-1)}\).

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References

  1. Chen B.-Y.: Minimal surfaces with constant Gauss curvature. Proc. Amer. Math. Soc. 34, 504–508 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen B.-Y.: Geometry of Submanifolds. Mercer Dekker, New York (1973)

    MATH  Google Scholar 

  3. Chen B.-Y.: Total Mean Curvature and Submanifolds of Finite Type. World Scientific, New Jersey (1984)

    MATH  Google Scholar 

  4. B.-Y. Chen, Riemannian submanifolds, Handbook of Differential Geometry, Vol. I, 187–418, North-Holland, Amsterdam, 2000 (eds. F. Dillen and L. Verstraelen).

  5. B.-Y. Chen, Complete classification of parallel spatial surfaces in pseudo-Riemannian space forms with arbitrary index and dimension, J. Geom. Phys. 60 (2010), in print.

  6. Chen B.-Y., Van der Veken J.: Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms. Tohoku Math. J. 61, 1–40 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dillen F.: Minimal immersions of surfaces into spheres. Arch. Math. (Basel) 49, 94–96 (1987)

    MATH  MathSciNet  Google Scholar 

  8. Ferus D.: Immersions with parallel second fundamental form. Math. Z. 140, 87–93 (1987)

    Article  MathSciNet  Google Scholar 

  9. Kenmotsu K.: Minimal surfaces with constant curvature in 4-dimensional space forms. Proc. Amer. Math. Soc. 89, 133–138 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  10. Magid M.A.: Isometric immersions of Lorentz space with parallel second fundamental forms. Tsukuba J. Math. 8, 31–54 (1984)

    MATH  MathSciNet  Google Scholar 

  11. O’Neill B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1982)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824–1027, USA

    Bang-Yen Chen

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  1. Bang-Yen Chen
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Correspondence to Bang-Yen Chen.

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Dedicated to Professor Tadashi Nagano on the occasion of his 80th birthday

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Chen, BY. A minimal immersion of the hyperbolic plane into the neutral pseudo-hyperbolic 4-space and its characterization. Arch. Math. 94, 291–299 (2010). https://doi.org/10.1007/s00013-009-0094-4

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  • DOI: https://doi.org/10.1007/s00013-009-0094-4

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