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On the uniqueness of the helicoid and Enneper's surface in the Lorentz-Minkowski space $ \mathbb{R}_1^3$

Authors: Isabel Fernandez and Francisco J. Lopez
Journal: Trans. Amer. Math. Soc. 363 (2011), 4603-4650
MSC (2000): Primary 53A10; Secondary 53C42, 53C50
Published electronically: April 20, 2011
MathSciNet review: 2806686
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Abstract: In this paper we deal with the uniqueness of the Lorentzian helicoid and Enneper's surface among properly embedded maximal surfaces with lightlike boundary of mirror symmetry in the Lorentz-Minkowski space $ \mathbb{R}_1^3.$

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  • 1. L. V. Ahlfors and L. Sario: Riemann surfaces. Princeton Univ. Press, Princeton, New Jersey, 1960. MR 0114911 (22:5729)
  • 2. R. Bartnik and L. Simon: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Comm. Math. Phys., Vol. 87 (1982/83), 131-152. MR 680653 (84j:58126)
  • 3. T. H. Colding and W. P. Minicozzi: The space of embedded minimal surfaces of fixed genus in a $ 3$-manifold IV; Locally simply connected. Ann. of Math., Vol. 160 (2004), 573-615. MR 2123933 (2006e:53013)
  • 4. K. Ecker: Area maximizing hypersurfaces in Minkowski space having an isolated singularity. Manuscripta Math., Vol. 56 (1986), 375-397. MR 860729 (88b:53075)
  • 5. E. Calabi: Examples of the Bernstein problem for some nonlinear equations. Proc. Symp. Pure Math., Vol. 15 (1970), 223-230. MR 0264210 (41:8806)
  • 6. S. Y. Cheng and S. T. Yau: Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Annals of Mathematics, Vol. 104 (1976), 407-419. MR 0431061 (55:4063)
  • 7. P. Collin, R. Kusner, W.H. Meeks III, and H. Rosenberg: The topology, geometry and conformal structure of properly embedded minimal surfaces. Journal of Differential Geometry, Vol. 67 (2004), no. 2, 377-393. MR 2153082 (2006j:53004)
  • 8. I. Fernández, F. J. López and R. Souam: The space of complete embedded maximal surfaces with isolated singularities in the $ 3$-dimensional Lorentz-Minkowski space $ {\mathbb{L}}^3.$ Math. Ann., Vol. 332 (2005), 605-643. MR 2181764 (2006h:58012)
  • 9. S. Fujimori, K. Saji, M. Umehara and K. Yamada: Singularities of maximal surfaces. Math. Z. 259 (2008), no. 4, 827-848. MR 2403743 (2009e:53009)
  • 10. H. Fujimoto: On the number of exceptional values of the Gauss map of minimal surfaces. J. Math. Soc. Japan, 40, 235-247 (1988). MR 930599 (89b:53013)
  • 11. H. Fujimoto: Modiffied defect relations for the Gauss map of minimal surfaces. J. Differential Geom. 29 (1989), 245-262. MR 982173 (89m:53012)
  • 12. A. Grigor'yan: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc 36 (1999), 135-249. MR 1659871 (99k:58195)
  • 13. A. A. Klyachin and V. M. Miklyukov: Traces of functions with spacelike graphs, and the extension problem under restriction on the gradient. Russian Acad. Sci. Sb. Math., Vol. 76 (1993), No. 2. MR 1186974 (93i:46060)
  • 14. O. Kobayashi: Maximal surfaces in the $ 3$-dimensional Minkowski space $ {\mathbb{L}}^3.$ Tokyo J. Math., Vol. 6 (1983), no. 2, 297-309. MR 732085 (85d:53003)
  • 15. O. Kobayashi: Maximal surfaces with conelike singularities. J. Math. Soc. Japan 36 (1984), no. 4, 609-617. MR 759417 (86d:53008)
  • 16. P. Li and J. Wang: Finiteness of disjoint minimal graphs. Math. R. Letters, Vol. 6 (2002), no. 5-6, 771-778. MR 1879819 (2002k:53012)
  • 17. F. J. López and J. Pérez: Parabolicity and Gauss map of minimal surfaces. Indiana Univ. Math. J. 52 (2003), no. 4, 1017-1026. MR 2001943 (2004f:53005)
  • 18. J. Pérez: Stable embedded minimal surfaces bounded by a straight line. Calculus of Variations and Partial Differential Equations 29 (2007), no. 2, 267-279. MR 2307776 (2008e:53018)
  • 19. J. Pérez, A. Ros: Properly embedded minimal surfaces with finite total curvature. ``The Global Theory of Minimal Surfaces in Flat Spaces'', Lecture Notes in Mathematics, Springer-Verlag, Vol. 1775 (2002), 15-66. MR 1901613
  • 20. J. E. Marsden and F. J. Tipler.: Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Phys. Rep., Vol. 66 (1980), no. 3, 109-139. MR 598585 (82d:83048)
  • 21. W. H. Meeks III and H. Rosenberg: The uniqueness of the helicoid. Ann. Math. 161 (2005), 727-758. MR 2153399 (2006f:53012)
  • 22. N. Quien: Plateau's problem in Minkowski space. Analysis 5 (1985), 43-60. MR 791491 (86m:53013)
  • 23. J. L. Schiff: Normal families. Universitext, Springer-Verlag, New York, 1993. MR 1211641 (94f:30046)
  • 24. J. Pérez: Parabolicity and minimal surfaces. Proceedings of the conference on global theory of minimal surfaces, Berkeley (2003).
  • 25. M. Umehara and K. Yamada: Maximal surfaces with singularities in Minkowski space. Hokkaido Mathematical Journal, Vol. 35 (2006), no. 1, pp. 13-40. MR 2225080 (2007a:53018)

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Additional Information

Isabel Fernandez
Affiliation: Departamento de Matematica Aplicada I, Facultad de Informática, Universidad de Sevilla, 41012, Sevilla, Spain

Francisco J. Lopez
Affiliation: Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

Keywords: Maximal surface, multigraph.
Received by editor(s): August 7, 2008
Received by editor(s) in revised form: June 9, 2009
Published electronically: April 20, 2011
Additional Notes: The first author’s research was partially supported by MCYT-FEDER research project MTM2007-64504, and Junta de Andalucia Grants P06-FQM-01642 and FQM325.
The second author’s research was partially supported by MCYT-FEDER research project MTM2007-61775 and Junta de Andalucia Grant P06-FQM-01642.
Article copyright: © Copyright 2011 American Mathematical Society

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