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Embedded isolated singularities of flat surfaces in hyperbolic 3-space

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Abstract.

We give a complete description of the flat surfaces in hyperbolic 3-space that are regularly embedded around an isolated singularity. Specifically, we show that there is a one-to-one explicit correspondence between this class and the class of regular analytic convex Jordan curves in the 2-sphere. Previously, the only known examples of such surfaces were rotational ones. To achieve this result, we first solve the geometric Cauchy problem for flat surfaces in hyperbolic 3-space.

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References

  1. Al\’ ıas, L.J., Chaves, R.M.B., Mira, P.: Björling problem for maximal surfaces in Lorentz-Minkowski space. Math. Proc. Cambridge Philos. Soc. 134, 289-316 (2003)

    Google Scholar 

  2. Bryant, R.L.: Surfaces of mean curvature one in hyperbolic space. Astérisque 154-155, 321-347 (1987)

    Google Scholar 

  3. Gálvez, J.A., Martínez, A., Milán, F.: Flat surfaces in the hyperbolic 3-space. Math. Ann. 316, 419-435 (2000)

    Article  Google Scholar 

  4. Gálvez, J.A., Martínez, A., Milán, F.: Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity. Trans. Amer. Math. Soc. 356, 3405-3428 (2004)

    Article  Google Scholar 

  5. Gálvez, J.A., Mira, P.: The Cauchy problem for the Liouville equation and Bryant surfaces. Adv. Math. (to appear) (http://arXiv.org/math.DG/0310092)

  6. Kokubu, M., Rossman, W., Saji, K., Umehara, M., Yamada, K.: Singularities of flat fronts in hyperbolic 3-space. Preprint (available at http://arXiv.org/math.DG/0401110)

  7. Kokubu, M., Umehara, M., Yamada, K.: Flat fronts in hyperbolic 3-space. Pacific J. Math. 216, 149-175 (2004)

    Google Scholar 

  8. Li, A.M., Simon, U., Zhao, G.S.: Global affine differential geometry of hypersurfaces (de Gruyter Expositions in Mathematics) Walter de Gruyter, Berlin New York 1993

  9. Mira, P.: Ph.D. Thesis, University of Murcia (2003)

  10. Roitman, P.: Flat surfaces in hyperbolic 3-space as normal surfaces to a congruence of geodesics. Preprint

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    José A. Gálvez

  2. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30203, Cartagena, Murcia, Spain

    Pablo Mira

Authors
  1. José A. Gálvez
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  2. Pablo Mira
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Corresponding author

Correspondence to José A. Gálvez.

Additional information

Received: 19 September 2004, Accepted: 15 November 2004, Published online: 22 December 2004

J.A. Gálvez was partially supported by MCYT-FEDER, Grant no. BFM2001-3318.

P. Mira was partially supported by MCYT, Grant no. BFM2001-2871 and CARM Programa Séneca, Grant no PI-3/00854/FS/01.

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Gálvez, J.A., Mira, P. Embedded isolated singularities of flat surfaces in hyperbolic 3-space. Calc. Var. 24, 239–260 (2005). https://doi.org/10.1007/s00526-004-0321-6

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  • DOI: https://doi.org/10.1007/s00526-004-0321-6

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