Zero mean curvature surfaces in $ {\mathbf{L}}^{3}$ containing a light-like line

Fujimori, S.; Kim, Y.W.; Koh, S.-E.; Rossman, W.; Shin, H.; Takahashi, H.; Umehara, M.; Yamada, K.; Yang, S.-D.

doi:10.1016/j.crma.2012.10.024

Differential Geometry

Zero mean curvature surfaces in

L^{3}

containing a light-like line
[Surfaces de courbure moyenne nulle dans

L^{3}

contenant des droites de type lumière]

Présenté par : the Editorial Board

Fujimori, S.¹ ; Kim, Y.W.² ; Koh, S.-E.³ ; Rossman, W.⁴ ; Shin, H.⁵ ; Takahashi, H.⁶ ; Umehara, M.⁷ ; Yamada, K.⁸ ; Yang, S.-D.²

¹ Department of Mathematics, Faculty of Science, Okayama University, Okayama 700-8530, Japan
² Department of Mathematics, Korea University, Seoul 136-701, Republic of Korea
³ Department of Mathematics, Konkuk University, Seoul 143-701, Republic of Korea
⁴ Department of Mathematics, Faculty of Science, Kobe University, Kobe 657-8501, Japan
⁵ Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea
⁶ Hakuho Girlsʼ High School, Yokohama 230-0074, Japan
⁷ Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan
⁸ Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

Comptes Rendus. Mathématique, Tome 350 (2012) no. 21-22, pp. 975-978.

Résumé
Abstract

Il est bien connu que les surfaces maximales de type espace et les surfaces minimales de type temps dans lʼespace $L^{3}$ de Lorentz–Minkowski de dimension 3 possèdent en général des singularités. Ces deux types sont caracterisés comme des surfaces de courbure moyenne nulle. La Note considère le cas où le lieu des singularités consiste en une droite de type lumière, cette situation nʼayant semble-t-il pas encore été analysée. Dans cette Note, nous donnons de nouveaux exemples de telles surfaces.

It is well known that space-like maximal surfaces and time-like minimal surfaces in Lorentz–Minkowski 3-space $L^{3}$ have singularities (i.e. points where the induced metric degenerates) in general. We are interested in the case where the singular set consists of a light-like line, since this case has not been analyzed before. In this Note, we give new examples of such surfaces.

Reçu le : 2012-05-12
Accepté le : 2012-10-25
Publié le : 2012-11-01

DOI : 10.1016/j.crma.2012.10.024

Affiliations des auteurs :

Fujimori, S. ¹ ; Kim, Y.W. ² ; Koh, S.-E. ³ ; Rossman, W. ⁴ ; Shin, H. ⁵ ; Takahashi, H. ⁶ ; Umehara, M. ⁷ ; Yamada, K. ⁸ ; Yang, S.-D. ²

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     title = {Zero mean curvature surfaces in $ {\mathbf{L}}^{3}$ containing a light-like line},
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Fujimori, S.; Kim, Y.W.; Koh, S.-E.; Rossman, W.; Shin, H.; Takahashi, H.; Umehara, M.; Yamada, K.; Yang, S.-D. Zero mean curvature surfaces in $ {\mathbf{L}}^{3}$ containing a light-like line. Comptes Rendus. Mathématique, Tome 350 (2012) no. 21-22, pp. 975-978. doi : 10.1016/j.crma.2012.10.024. http://www.numdam.org/articles/10.1016/j.crma.2012.10.024/

Bibliographie
Cité par

[1] Estudillo, F.J.M.; Romero, A. Generalized maximal surfaces in Lorentz–Minkowski space $L^{3}$ , Math. Proc. Cambridge Philos. Soc., Volume 111 (1992), pp. 515-524

[2] 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 which change type across a light-like line, preprint.

[3] Fujimori, S.; Rossman, W.; Umehara, M.; Yamada, K.; Yang, S.-D. New maximal surfaces in Minkowski 3-space with arbitrary genus and their cousins in de Sitter 3-space, Results Math., Volume 56 (2009), pp. 41-82

[4] Gu, C. The extremal surfaces in the 3-dimensional Minkowski space, Acta Math. Sinica, Volume 1 (1985), pp. 173-180

[5] Inoguchi, J.; Toda, M. Timelike minimal surfaces via loop groups, Acta Appl. Math., Volume 83 (2004), pp. 313-335

[6] Kim, Y.W.; Yang, S.-D. A family of maximal surfaces in Lorentz–Minkowski three-space, Proc. Amer. Math. Soc., Volume 134 (2006), pp. 3379-3390

[7] Kim, Y.W.; Yang, S.-D. Prescribing singularities of maximal surfaces via a singular Björling representation formula, J. Geom. Phys., Volume 57 (2007), pp. 2167-2177

[8] Kim, Y.W.; Koh, S.-E.; Shin, H.; Yang, S.-D. Spacelike maximal surfaces, timelike minimal surfaces, and Björling representation formulae, J. Korean Math. Soc., Volume 48 (2011), pp. 1083-1100

[9] Klyachin, V.A. Zero mean curvature surfaces of mixed type in Minkowski space, Izv. Math., Volume 67 (2003), pp. 209-224

[10] Kobayashi, O. Maximal surfaces in the 3-dimensional Minkowski space $L^{3}$ , Tokyo J. Math., Volume 6 (1983), pp. 297-309

[11] Umehara, M.; Yamada, K. Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J., Volume 35 (2006), pp. 13-40

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