Mixed type surfaces with bounded mean curvature in 3-dimensional space-times

Honda, Atsufumi; Koiso, Miyuki; Kokubu, Masatoshi; Umehara, Masaaki; Yamada, Kotaro

doi:10.1016/j.difgeo.2017.03.009

Cited by 18 publications

(17 citation statements)

References 6 publications

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“…Surfaces in the class Y r are investigated in [10], and an entire graph in Y r which is not a ZMC-surface was given. In this section, we shall show a general existence result of surfaces in the class Y ω .…”

Section: Preliminariesmentioning

confidence: 99%

Surfaces With Light-Like Points In Lorentz-Minkowski 3-Space With Applications

Umehara

Yamada

2017

Lorentzian Geometry and Related Topics

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With several concrete examples of zero mean curvature surfaces in R 3 1 containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the first fundamental form of a surface degenerates is said to be light-like. We also show a theorem on a property of light-like points of a surface in R 3 1 whose mean curvature vector is smoothly extendable. This explains why such surfaces will contain a light-like line when they do not change causal types. Moreover, several applications of these two results are given.

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Section: Preliminariesmentioning

confidence: 99%

Surfaces With Light-Like Points In Lorentz-Minkowski 3-Space With Applications

Umehara

Yamada

2017

Lorentzian Geometry and Related Topics

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“…The type-change of causal characters of mixed type ZMC surfaces corresponds to the change of a stream function being from subsonic to supersonic (for more precise, see [6]). We remark that, in [16], it was proved that there do not exist mixed type surfaces with non-zero constant mean curvature (see also [44,45]).…”

Section: Introductionmentioning

confidence: 94%

“…1 In the case of wave fronts, singular points of the first kind and the second kind were introduced in [28]. 6 In our terminology, [16,Proposition 3.5] can be interpreted as follows.…”

Section: 1mentioning

confidence: 99%

Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifolds

Honda

Saji

Teramoto

2020

Advances in Mathematics

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A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the behavior of Gaussian curvature at a non-degenerate lightlike point of a mixed type surface. To characterize the boundedness of Gaussian curvature at a non-degenerate lightlike points, we introduce several fundamental invariants along non-degenerate lightlike points, such as the lightlike singular curvature and the lightlike normal curvature. Moreover, using the results by Pelletier and Steller, we obtain the Gauss-Bonnet type formula for mixed type surfaces with bounded Gaussian curvature.

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“…By this theorem, we can not expect the existence of CMC surfaces of mixed type. In fact, in [17] with Kokubu, Umehara and Yamada, the first and second authors have proved that there do not exist (connected) CMC surfaces of mixed type.…”

Section: Introductionmentioning

confidence: 99%

Fold singularities on spacelike CMC surfaces in Lorentz-Minkowski space

Honda

Koiso²,

Saji

2018

Hokkaido Math. J.

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Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular points on spacelike surfaces with non-zero constant mean curvature (spacelike CMC surfaces). We prove that spacelike CMC surfaces do not admit fold singular points. Moreover, we show that the singular point set of any conjugate CMC surface of a spacelike Delaunay surface with conelike singular points consists of (2, 5)-cuspidal edges.

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Mixed type surfaces with bounded mean curvature in 3-dimensional space-times

Cited by 18 publications

References 6 publications

Surfaces With Light-Like Points In Lorentz-Minkowski 3-Space With Applications

Surfaces With Light-Like Points In Lorentz-Minkowski 3-Space With Applications

Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifolds

Fold singularities on spacelike CMC surfaces in Lorentz-Minkowski space

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