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Singularities of spacelike constant mean curvature surfaces in Lorentz–Minkowski space

Published online by Cambridge University Press:  15 March 2011

DAVID BRANDER
DAVID BRANDER*
Affiliation:
Department of Mathematics, Matematiktorvet, Building 303 S, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark. e-mail: [email protected]
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Abstract

We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space L3. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve f0(x), and a real analytic null vector field v(x) parallel to the tangent field of f0, find a conformally parameterized (generalized) CMC H surface in L3 which contains this curve as a singular set and such that the partial derivatives fx and fy are given by df0/dx and v along the curve. Within the class of generalized surfaces considered, the solution is unique and we give a formula for the generalized Weierstrass data for this surface. This gives a framework for studying the singularities of non-maximal CMC surfaces in L3. We use this to find the Björling data – and holomorphic potentials – which characterize cuspidal edge, swallowtail and cuspidal cross cap singularities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 3 , May 2011 , pp. 527 - 556
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Akutagawa, K. and Nishikawa, S.The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space. Tohoku Math. J. (2) 42 (1990), 67–82.CrossRefGoogle Scholar
[2]Alías, L. J., Chaves, R. M. B. and Mira, P.Björling problem for maximal surfaces in Lorentz-Minkowski space. Math. Proc. Camb Phil. Soc. 134 (2003), 289–316.CrossRefGoogle Scholar
[3]Arnold, V. I. Singularities of caustics and wave fronts Math. Appl. (Soviet Series). vol. 62 (Kluwer Academic Publishers Group 1990).Google Scholar
[4]Brander, D. and Dorfmeister, J. F.The Björling problem for non-minimal constant mean curvature surfaces. Comm. Anal. Geom. 18 (2010), 171–194.CrossRefGoogle Scholar
[5]Brander, D., Rossman, W. and Schmitt, N.Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods. Adv. Math. 223 (2010), 949–986.CrossRefGoogle Scholar
[6]Dierkes, U., Hildebrandt, S., Küster, A. and Wohlrab, O.Minimal surfaces. I. Boundary value problems. Grundlehren der Mathematischen Wissenschaften. vol. 295 (Springer-Verlag, 1992).CrossRefGoogle Scholar
[7]Dorfmeister, J., Pedit, F. and Wu, H.Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6 (1998), 633–668.CrossRefGoogle Scholar
[8]Fernandez, I. and Lopez, F. J.Periodic maximal surfaces in the Lorentz-Minkowski space L 3. Math. Z. 256 (2007), 573–601.Google Scholar
[9]Fernandez, I., Lopez, F. J. and Souam, R.The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space. Math. Ann. 332 (2005), 605–643.CrossRefGoogle Scholar
[10]Fernandez, I., Lopez, F. J. and Souam, R.The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz-Minkowski space L 3. Manuscripta Math. 122 (2007), 573–601.CrossRefGoogle Scholar
[11]Fujimori, S., Saji, K., Umehara, M. and Yamada, K.Singularities of maximal surfaces. Math. Z. 259 (2008), 827–848.Google Scholar
[12]Inoguchi, J.Surfaces in Minkowski 3-space and harmonic maps. Harmonic morphisms, harmonic maps, and related topics (Brest, 1997), 249–270. Chapman & Hall/CRC Res. Notes Math. 413, (Chapman & Hall/CRC, 2000).Google Scholar
[13]Ishikawa, G. and Machida, Y.Singularities of improper affine spheres and surfaces of constant Gaussian curvature. Internat. J. Math. 17 (2006), 269–293.CrossRefGoogle Scholar
[14]Kenmotsu, K.Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann. 245 (1979), 89–99.CrossRefGoogle Scholar
[15]Kim, Y. W. and Yang, S. D.Prescribing singularities of maximal surfaces via a singular Björling representation formula. J. Geom. Phys. 57 (2007), 2167–2177.Google Scholar
[16]Kokubu, M., Rossman, W., Saji, K., Umehara, M. and Yamada, K.Singularities of flat fronts in hyperbolic space. Pacific J. Math. 221 (2005), 303–351.CrossRefGoogle Scholar
[17]Saji, K., Umehara, M. and Yamada, K.The geometry of fronts. Ann. of Math. (2) 169 (2009), 491–529.CrossRefGoogle Scholar
[18]Umeda, Y.Constant-mean-curvature surfaces with singularities in Minkowski 3-space. Experiment. Math. 18 (2009), 311–323.CrossRefGoogle Scholar
[19]Umehara, M. and Yamada, K.Maximal surfaces with singularities in Minkowski space. Hokkaido Math. J. 35 (2006), 13–40.Google Scholar
[20]Whitney, H.The singularities of a smooth n-manifold in (2n-1)-space. Ann. Math. 45 (1944), 247–293.CrossRefGoogle Scholar