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Generalized maximal surfaces in Lorentz–Minkowski space L3

Published online by Cambridge University Press:  24 October 2008

Francisco J. M. Estudillo  and
Alfonso Romero
Francisco J. M. Estudillo
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias Económicas y Empresariales (ETEA), Universidad de Córdoba, 14004-Córdoba, Spain
Alfonso Romero
Affiliation:
Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain
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Extract

In this paper we carry out a systematic study of generalized maximal surfaces in Lorentz–Minkowski space L3, with emphasis on their branch points. Roughly speaking, such a surface is given by a conformal mapping from a Riemann surface S in L3. In the last years, several authors [1, 2, 5, 6] have dealt with regular maximal surfaces in L3, i.e. with isometric immersions, with zero mean curvature, of Riemannian 2-manifolds M in L3. So, the term ‘regular’ means free of branch points. As in the minimal case, a conformal structure is naturally induced on M, which becomes a Riemann surface S. The corresponding isometric immersion is then conformal on S, and it does not have any singular points on S (i.e. points on which the differential of the mapping is not one-to-one). This is the way in which generalized maximal surfaces include regular ones. Moreover, branch points are the singular points of the conformal mapping on S. Whereas branch points of generalized minimal surfaces are isolated, we shall show in Section 2 that, in addition to isolated branch points, a generalized maximal surface in L3. may have non-isolated ones, in fact they constitute a 1-dimensional submanifold in a certain open subset of S (see Section 2). So our purpose is two-fold, firstly to study and explain in detail the branch points, and secondly to state several geometric results involving prescribed behaviour of those points on the surface.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 3 , May 1992 , pp. 515 - 524
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Calabi, E.. Examples of Bernstein problems for some non-linear equations. Proc. Sympos. Pure Math. 15 (1970), 223–230.CrossRefGoogle Scholar
[2]Chaohao, G.. The extremal surfaces in the 3-dimensional Minkowski space. Acta Math. Sinica 1 (1985), 173–180.CrossRefGoogle Scholar
[3]Fujimoto, H.. On the number of exceptional values of the Gauss map of minimal surfaces. J. Math. Soc. Japan 40 (1988), 235–247.CrossRefGoogle Scholar
[4]Huber, A.. On subharmonic functions and differential geometry in the large. Comment. Math. Helv. 32 (1957), 13–72.CrossRefGoogle Scholar
[5]Kobayashi, O.. Maximal surfaces in the 3-dimensional Minkowski space L3. Tokyo J. Math. 6 (1983), 297–309.CrossRefGoogle Scholar
[6]McNertney, L. V.. One-parameter families of surfaces with constant curvature in Lorentz 3-space. Ph.D. thesis, Brown University (1980).Google Scholar
[7]Osserman, R.. A Survey of Minimal Surfaces, 2nd edition (Dover, 1986).Google Scholar
[8]Rosenberg, H. and Toubiana, E.. Complete minimal surfaces and minimal herisson. J. Differential Geom. 28 (1988), 115–132.CrossRefGoogle Scholar