×

Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space. (English) Zbl 1441.53005

Summary: Consider a surface \(S\) immersed in the Lorentz-Minkowski 3-space \(\mathbb{R}^3_1\). A complete light-like line in \(\mathbb{R}^3_1\) is called an entire null line on \(S^3_1\) if it lies on \(S\) and consists only of 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 \(\mathbb{R}^2 \), 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.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
35M10 PDEs of mixed type
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] 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. · Zbl 1409.53010 · doi:10.2206/kyushujm.71.211
[2] 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. · Zbl 1417.53010 · doi:10.1007/s12044-019-0479-7
[3] 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.). · Zbl 1433.53017
[4] 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. · Zbl 1433.53017
[5] 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. · Zbl 0352.53021 · doi:10.2307/1970963
[6] 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. · Zbl 1257.53090
[7] 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. · Zbl 1319.53008
[8] 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. · Zbl 1414.53052
[9] 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. · Zbl 1076.53015 · doi:10.1070/IM2003v067n02ABEH000425
[10] O. Kobayashi, Maximal surfaces in the 3-dimensional Minkowski space \(L^3\), Tokyo J. Math. 6 (1983), no. 2, 297-309. · Zbl 0535.53052
[11] M. Umehara and K. Yamada, Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J. 35 (2006), no. 1, 13-40. · Zbl 1109.53016 · doi:10.14492/hokmj/1285766302
[12] 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. · Zbl 1402.53007
[13] M. Umehara and K. Yamada, Hypersurfaces with light-like points in a Lorentzian manifold, J. Geom. Anal. 29 (2019), 3405-3437. · Zbl 1430.53009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.