Spacelike Maximal Surfaces, Timelike Minimal Surfaces, and Björling Representation Formulae

Kim, Young‐Wook; Koh, Sung-Eun; Shin, Hyeonjoon; Yang, Seong-Deog

doi:10.4134/jkms.2011.48.5.1083

Cited by 42 publications

(52 citation statements)

References 13 publications

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“…Non-trivial minimal surfaces in L 3 are also available in the literature (see e.g. [35][36][37] for Lorentzian helicoids and catenoids). However, these are embedded in a non-trivial way in the time-like direction and hence most of these surfaces do not preserve a one parameter family of isometries of the ambient space-time.…”

Section: Discussionmentioning

confidence: 99%

“…This implies that the only non-vanishing component of the Ricci tensor is 37) and that the Ricci scalar is zero,…”

Section: Plane Wavesmentioning

confidence: 99%

“…Minimal surfaces in Lorentzian space-times L (D) have also been considered in the mathematics literature and some examples of minimal surfaces are known (see e.g. [35][36][37] for a selection of minimal surfaces). However, as we will explain in section 4, these are not of use for the purposes of this work.…”

Section: Jhep07(2015)156mentioning

confidence: 99%

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Blackfolds, plane waves and minimal surfaces

Armas

Blau

2015

J. High Energ. Phys.

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Minimal surfaces in Euclidean space provide examples of possible non-compact horizon geometries and topologies in asymptotically flat space-time. On the other hand, the existence of limiting surfaces in the space-time provides a simple mechanism for making these configurations compact. Limiting surfaces appear naturally in a given space-time by making minimal surfaces rotate but they are also inherent to plane wave or de Sitter spacetimes in which case minimal surfaces can be static and compact. We use the blackfold approach in order to scan for possible black hole horizon geometries and topologies in asymptotically flat, plane wave and de Sitter space-times. In the process we uncover several new configurations, such as black helicoids and catenoids, some of which have an asymptotically flat counterpart. In particular, we find that the ultraspinning regime of singly-spinning Myers-Perry black holes, described in terms of the simplest minimal surface (the plane), can be obtained as a limit of a black helicoid, suggesting that these two families of black holes are connected. We also show that minimal surfaces embedded in spheres rather than Euclidean space can be used to construct static compact horizons in asymptotically de Sitter space-times.

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Section: Discussionmentioning

confidence: 99%

“…This implies that the only non-vanishing component of the Ricci tensor is 37) and that the Ricci scalar is zero,…”

Section: Plane Wavesmentioning

confidence: 99%

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Blackfolds, plane waves and minimal surfaces

Armas

Blau

2015

J. High Energ. Phys.

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“…The spacelike helicoids and the timelike helicoids match analytically across their folded singularities, which is a general property of folded singularities of analytic surfaces with zero mean curvature [1,2].…”

Section: Preliminariesmentioning

confidence: 98%

Minimal harmonic graphs and their Lorentzian cousins

Kim

Lee

Yang

2009

Journal of Mathematical Analysis and Applications

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Motivated by the observation that the only surface which is locally a graph of a harmonic function and is also a minimal surface in E 3 is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L 3 as the nontrivial zero mean curvature surfaces which also satisfy the harmonic equation, the wave equation, and a degenerate equation which is derived from the harmonic equation or the wave equation. This elementary and analytic result shows that the change of the roles of dependent and independent variables may be useful in solving differential equations.

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“…Moreover, these surfaces often can be real analytically extended to ZMC surfaces which have more than one causal character. For example, motivated by the works [8,9,10,12,13], in [6], Fujimori et al showed that any maxface which has non-degenerate fold-singularities [6, Definition 2.13] is extended real analytically to a timelike minimal surface. In this extension, the set of fold singularities of the maxface forms a non-degenerate null curve on a ZMC surface, where a non-degenerate null curve is a regular curve whose velocity vector field is lightlike and not proportional to its acceleration vector field everywhere.…”

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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Spacelike Maximal Surfaces, Timelike Minimal Surfaces, and Björling Representation Formulae

Cited by 42 publications

References 13 publications

Blackfolds, plane waves and minimal surfaces

Blackfolds, plane waves and minimal surfaces

Minimal harmonic graphs and their Lorentzian cousins

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

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