Proceedings of the Japan Academy, Series A, Mathematical Sciences

Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space

Shintaro Akamine, Masaaki Umehara, and Kotaro Yamada

Full-text: Open access

PDF File (263 KB)

Abstract

Consider a surface $S$ immersed in the Lorentz-Minkowski 3-space $\mathbf{R}^{3}_{1}$. A complete light-like line in $\mathbf{R}^{3}_{1}$ is called an \textit{entire null line} on the surface $S$ in $\mathbf{R}^{3}_{1}$ if it lies on $S$ and consists of only null points with respect to the induced metric. In this paper, we show the existence of embedded space-like maximal graphs containing entire null lines. If such a graph is defined on a convex domain in $\mathbf{R}^{2}$, then it must be contained in a light-like plane (cf. Remark~3.3). Our example is critical in the sense that it is defined on a certain non-convex domain.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 95, Number 9 (2019), 97-102.

Dates
First available in Project Euclid: 1 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.pja/1572595222

Digital Object Identifier
doi:10.3792/pjaa.95.97

Mathematical Reviews number (MathSciNet)
MR4026357

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53B30: Lorentz metrics, indefinite metrics
Secondary: 35M10: Equations of mixed type

Keywords
Maximal surface type change zero mean curvature Lorentz-Minkowski space

Citation

Akamine, Shintaro; Umehara, Masaaki; Yamada, Kotaro. Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space. Proc. Japan Acad. Ser. A Math. Sci. 95 (2019), no. 9, 97--102. doi:10.3792/pjaa.95.97. https://projecteuclid.org/euclid.pja/1572595222


Export citation

References

  • S. Akamine, Causal characters of zero mean curvature surfaces of Riemann type in the Lorentz-Minkowski 3-space, Kyushu J. Math. 71 (2017), no. 2, 211–249.
    Zentralblatt MATH: 1409.53010
    Digital Object Identifier: doi:10.2206/kyushujm.71.211
  • S. Akamine and R. K. Singh, Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born-Infeld equation, Proc. Indian Acad. Sci. Math. Sci. 129 (2019), no. 3, Art. 35.
    Zentralblatt MATH: 1417.53010
    Digital Object Identifier: doi:10.1007/s12044-019-0479-7
  • S. Akamine, M. Umehara and K. Yamada, Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality, (to appear in Proc. Amer. Math. Soc.).
  • S. Akamine, A. Honda, M. Umehara and K. Yamada, Bernstein-type theorem for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski space, arXiv:1907.01754.
    arXiv: 1907.01754
  • S. Y. Cheng and S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2) 104 (1976), no. 3, 407–419.
    Zentralblatt MATH: 0352.53021
    Digital Object Identifier: doi:10.2307/1970963
  • S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin, H. Takahashi, M. Umehara, K. Yamada and S.-D. Yang, Zero mean curvature surfaces in $\mathbf{L}^{3}$ containing a light-like line, C. R. Math. Acad. Sci. Paris 350 (2012), no. 21–22, 975–978.
  • S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin, M. Umehara, K. Yamada and S.-D. Yang, Zero mean curvature surfaces in Lorentz-Minkowski 3-space which change type across a light-like line, Osaka J. Math. 52 (2015), no. 1, 285–297, Erratum: Osaka J. Math. 53 (2016), no. 1, 289–292.
    Zentralblatt MATH: 1319.53008
    Project Euclid: euclid.ojm/1427202882
  • K. Hashimoto and S. Kato, Bicomplex extensions of zero mean curvature surfaces in $\mathbf{R}^{2,1}$ and $\mathbf{R}^{2,2}$, J. Geom. Phys. 138 (2019), 223–240.
  • V. A. Klyachin, Zero mean curvature surfaces of mixed type in Minkowski space, Izv. Math. 67 (2003), no. 2, 209–224; translated from Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 2, 5–20.
    Zentralblatt MATH: 1076.53015
    Digital Object Identifier: doi:10.1070/IM2003v067n02ABEH000425
  • O. Kobayashi, Maximal surfaces in the 3-dimensional Minkowski space $L^{3}$, Tokyo J. Math. 6 (1983), no. 2, 297–309.
  • M. Umehara and K. Yamada, Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J. 35 (2006), no. 1, 13–40.
    Zentralblatt MATH: 1109.53016
    Digital Object Identifier: doi:10.14492/hokmj/1285766302
    Project Euclid: euclid.hokmj/1285766302
  • M. Umehara and K. Yamada, Surfaces with light-like points in Lorentz-Minkowski 3-space with applications, in Lorentzian geometry and related topics, Springer Proc. Math. Stat., 211, Springer, Cham, 2017, pp. 253–273.
  • M. Umehara and K. Yamada, Hypersurfaces with light-like points in a Lorentzian manifold, J. Geom. Anal. 29 (2019), 3405–3437.

The Japan Academy

Proceedings of the Japan Academy, Series A, Mathematical Sciences

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more