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

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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


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References

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