Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-12T05:29:34.859Z Has data issue: false hasContentIssue false
  • English
  • Français

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: D.Brander@mat.dtu.dk
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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