Zero mean curvature surfaces in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="bold">L</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> containing a light-like line

Fujimori, Shoichi; Kim, Y. W.; Koh, Sung-Eun; Rossman, Wayne; Shin, Hyeonjoon; Takahashi, Hiroaki; Umehara, Masaaki; Yamada, Kotaro; Yang, Seong-Deog

doi:10.1016/j.crma.2012.10.024

Cited by 30 publications

(78 citation statements)

References 10 publications

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“…For the case of extremal surfaces of mixed type, Gu gave the construction of mixed type extremal surface using the geometrical method [16][17][18][19]. For lightlike case, we can see [14,20,23] and references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global weak and smooth solutions of the equation for timelike extremal surface in Minkowski space

Lai

Liu

2015

Journal of Mathematical Analysis and Applications

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The time-like extremal surface is an interesting model which may be viewed as simple but nontrivial examples of membrane in field theory. In this paper, we first consider the global weak solutions to the timelike extremal surface in Minkowski space R 1+(1+n) . Furthermore, we can get global smooth solutions to the timelike extremal hypersurface for initial data sufficiently close to the arbitrary time-like plane a 0 t + a 1 x 1 + · · · + a n x n + b = 0 in Minkowski space R 1+(n+1) , but not in the initial plane t = 0.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Global weak and smooth solutions of the equation for timelike extremal surface in Minkowski space

Lai

Liu

2015

Journal of Mathematical Analysis and Applications

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“…As shown in [6] and [12], by a homothetic change one can normalize µ to be −1, 0 or 1. In fact, as shown in [6],…”

Section: Zmc-surfaces With Null Linesmentioning

confidence: 99%

“…Klyachin [9] showed the following fact: This fact was generalized to a much wider class of surfaces, including real analytic constant mean curvature surfaces in R 3 1 , see [12,13]. Although there are properly embedded time-like ZMC-surfaces with an entire null line (see [3, Examples 2.2 and 2.3]), each of all examples of ZMC-surfaces with space-like points given in [1,2,6,10] containing an entire null line L has at least one cone-like singular point on L (see Fig. 2).…”

Section: Introductionmentioning

confidence: 99%

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

Akamine¹,

Umehara²,

Yamada³

2019

Proc. Japan Acad. Ser. A Math. Sci.

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Consider a surface S immersed in the Lorentz-Minkowski 3-space R 3 1 . A complete light-like line in R 3 1 is called an entire null line on the surface S in 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 R 2 , then it must be 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.

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“…Table 1 shows the explicit solutions α(y) of (5.1), and Table 2 shows examples of ZMC surfaces containing a lightlike line. See the references [4,5,6] for details.…”

Section: Characteristic Of Zmc Surfaces Containing a Lightlike Linementioning

confidence: 99%

“…On the other hand, there are no similar results for ZMC surfaces whose lightlike parts are straight lines, which degenerate everywhere. However, several ZMC surfaces which have exactly two causal characters and their lightlike parts are straight lines were constructed in [4], and there is a one-parameter family of ZMC surfaces which change their causal character from spacelike to timelike across a lightlike line constructed in [5]. These surfaces are the only known examples of ZMC surfaces which change their causal characters from spacelike to timelike across a lightlike line.…”

Section: Introductionmentioning

confidence: 99%

Causal Characters of Zero Mean Curvature Surfaces of Riemann Type in the Lorentz-Minkowski 3-Space

Akamine

2017

Kyushu J. Math.

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Abstract. A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann type according to their causal characters, and as a corollary, we prove that if a zero mean curvature surface of Riemann type has exactly two causal characters, then the lightlike part of the surface is a part of a straight line.

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Zero mean curvature surfaces in L3 containing a light-like line

Cited by 30 publications

References 10 publications

Global weak and smooth solutions of the equation for timelike extremal surface in Minkowski space

Global weak and smooth solutions of the equation for timelike extremal surface in Minkowski space

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

Causal Characters of Zero Mean Curvature Surfaces of Riemann Type in the Lorentz-Minkowski 3-Space

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