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

Akamine, Shintaro

doi:10.2206/kyushujm.71.211

Cited by 14 publications

(33 citation statements)

References 17 publications

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“…Thus, we can conclude that ψ(Ω) lies in a light-like plane. In [1], the first author constructed several ZMC-surfaces foliated by circles and at most countably many straight lines. At the end of this paper, we pick up two important examples of them which contain degenerate light-like points.…”

Section: Proof Of Theorem Amentioning

confidence: 99%

“…At the end of this paper, we pick up two important examples of them which contain degenerate light-like points. (In [1], these two examples are not precisely indicated. Here we show their explicit parametrization and embeddedness.)…”

Section: Proof Of Theorem Amentioning

confidence: 99%

“…On the other hand, there are several examples of ZMC-surfaces with degenerate light-like points (cf. [1,2,10,14]). Moreover, a local general existence theorem for maximal surfaces with degenerate light-like points is given in [21].…”

Section: Introductionmentioning

confidence: 99%

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Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

Akamine

Umehara

Yamada

2020

Proc. Amer. Math. Soc. Ser. B

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Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space L 3 which admits only spacelike points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space E 3 and maximal surfaces in Lorentz-Minkowski space L 3 , we give an improvement of this Bernstein-type theorem. More precisely, we show that a zero mean curvature entire graph in L 3 which does not admit time-like points (namely, a graph consists of only space-like and light-like points) is a plane.

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Section: Proof Of Theorem Amentioning

confidence: 99%

Section: Proof Of Theorem Amentioning

confidence: 99%

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Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

Akamine

Umehara

Yamada

2020

Proc. Amer. Math. Soc. Ser. B

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“…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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“…It is known that there is a duality between solutions to (1) and spacelike solutions to (2) called the Calabi's correspondence [4] as follows. Let z = f (x, y) be a minimal graph over a simply-connected domain.…”

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confidence: 99%

Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born–Infeld equation

Akamine

Singh

2019

Proc Math Sci

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A. In this paper we investigate relations between solutions to the minimal surface equation in Euclidean 3-space E 3 , the zero mean curvature equation in Lorentz-Minkowski 3-space L 3 and the Born-Infeld equation under Wick rotations. We prove that the existence conditions of real solutions and imaginary solutions after Wick rotations are written by symmetries of solutions, and reveal how real and imaginary solutions are transformed under Wick rotations. We also give a transformation theory for zero mean curvature surfaces containing lightlike lines with some symmetries. As an application, we give new correspondences among some solutions to the above equations by using the noncommutativity between Wick rotations and isometries in the ambient space.2010 Mathematics Subject Classification. Primary 53A10; Secondary 58J72, 53B30.

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

Cited by 14 publications

References 17 publications

Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

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

Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born–Infeld equation

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