# Geometry of Timelike Minimal Surfaces and Null Curves

## Abstract

In this chapter, we investigate the behavior of the Gaussian curvature of timelike minimal surfaces with or without singular points in the 3-dimensional Lorentz–Minkowski space. For timelike minimal surfaces without singular points, we prove that the sign of the Gaussian curvature, which corresponds to diagonalizability of the shape operator, of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also determine the behavior of the Gaussian curvature near cuspidal edges, swallowtails, and cuspidal cross caps on timelike minimal surfaces. We show that there are no umbilic points near cuspidal edges on a timelike minimal surface. Near swallowtails, we show that the sign of the Gaussian curvature is negative, that is, we can take always real principal curvatures near swallowtails. Near cuspidal cross caps, we also show that the sign of the Gaussian curvature is positive, that is, we can take only complex principal curvatures near cuspidal cross caps.

## Keywords

Lorentz–Minkowski space Timelike minimal surface Gaussian curvature Singularity## Notes

### Acknowledgements

The author would like to express his gratitude to Professor Miyuki Koiso for her helpful advices and suggestions. He also thanks the referee for valuable comments. This work was supported by Grant-in-Aid for JSPS Fellows Number 15J06677.

## References

- 1.S. Akamine,
*Behavior of the Gaussian Curvature of Timelike Minimal Surfaces with Singularities*, submitted, arXiv:1701.00238 - 2.G. Bellettini, J. Hoppe, M. Novaga, G. Orlandi, Closure and convexity results for closed relativistic strings. Complex Anal. Oper. Theory
**4**, 473–496 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 3.S. Chandrasekhar, Surface tension of liquid crystals. Mol. Cryst.
**2**, 71–80 (1966)CrossRefGoogle Scholar - 4.J.N. Clelland, Totally quasi-umbilic timelike surfaces in \(\mathbb{R}^{1,2}\). Asian J. Math.
**16**, 189–208 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 5.S. Fujimori, Y.W. Kim, S.-E. Koh, W. Rossman, H. Shin, M. Umehara, K. Yamada, S.-D. Yang, Zero mean curvature surfaces in Lorentz-Minkowski \(3\)-space and \(2\)-dimensional fluid mechanics. Math. J. Okayama Univ.
**57**, 173–200 (2015)MathSciNetzbMATHGoogle Scholar - 6.S. Fujimori, K. Saji, M. Umehara, K. Yamada, Singularities of maximal surfaces. Math. Z.
**259**, 827–848 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 7.A. Honda, M. Koiso, Y. Tanaka, Non-convex anisotropic surface energy and zero mean curvature surfaces in the Lorentz-Minkowski space. J. Math-for-Ind.
**5**, 73–82 (2013)MathSciNetzbMATHGoogle Scholar - 8.J. Inoguchi, M. Toda, Timelike minimal surfaces via loop groups. Acta Appl. Math.
**63**, 313–355 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Y.W. Kim, S.-E. Koh, H. Shin, S.-D. Yang, Spacelike maximal surfaces, timelike minimal surfaces, and Björling representation formulae. J. Korean Math. Soc.
**48**, 1083–1100 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 10.J. Konderak, A Weierstrass representation theorem for Lorentz surfaces. Complex Var. Theory Appl.
**50**(5), 319–332 (2005)MathSciNetzbMATHGoogle Scholar - 11.S. Lee, Weierstrass representation for timelike minimal surfaces in Minkowski 3-space. Commun. Math. Anal., Conf. 01, 11–19 (2008)Google Scholar
- 12.R. López, Differential Geometry of curves and surfaces in Lorentz-Minkowski space. Int. Electron. J. Geom.
**7**, 44–107 (2014)MathSciNetzbMATHGoogle Scholar - 13.L. McNertney, One-parameter families of surfaces with constant curvature in Lorentz \(3\)-space. Ph.D. thesis, Brown University, 1980Google Scholar
- 14.T.K. Milnor, Entire timelike minimal surfaces in \(\mathbb{E}^{3,1}\). Mich. Math. J.
**37**, 163–177 (1990)MathSciNetCrossRefGoogle Scholar - 15.Z. Olszak, A note about the torsion of null curves in the \(3\)-dimensional Minkowski spacetime and the Schwarzian derivative. Filomat
**29**, 553–561 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 16.H. Takahashi, Timelike minimal surfaces with singularities in three-dimensional spacetime. Master thesis, Osaka University, 2012, JapaneseGoogle Scholar
- 17.J.E. Taylor, Crystalline variational problems. Bull. Amer. Math. Soc.
**84**, 568–588 (1978)MathSciNetCrossRefzbMATHGoogle Scholar - 18.M. Umehara, K. Yamada, Maximal surfaces with singularities in Minkowski space. Hokkaido Math. J.
**35**, 13–40 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 19.T. Weinstein,
*An Introduction to Lorentz Surfaces*. De Gruyter Exposition in Mathematics, vol. 22 (Walter de Gruyter, Berlin, 1996)Google Scholar