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RETRACTED ARTICLE: Extremal surface with the light-like line in Minkowski space \(R^{1+(1+1)}\)
- Ruihua Gao^{1},
- Faxing Wang^{2},
- Xiaodan Zhang^{3} and
- Yuguang Wang^{4}Email authorView ORCID ID profile
- Received: 10 December 2016
- Accepted: 4 April 2017
- Published: 20 April 2017
The Retraction Note to this article has been published in Boundary Value Problems 2018 2018:69
Abstract
In this paper, firstly we will give the global construction of the mixed type extremal surface in Minkowski space along the analytic light-like line. Furthermore, we construct simply the local existence of extremal surface along a single light-like line.
Keywords
- classical solution
- extremal surface
- mixed type equation
MSC
- 35M10
- 35B65
1 Introduction
It is important to study the extremal surfaces in the theory of elementary particle physics, and it has also drawn attentions by mathematicians in geometrical analysis. In Minkowski space, the extremal surfaces include space-like type, time-like type, light-like type and mixed type. For time-like case, Milnor gave entire time-like minimal surfaces in the three-dimensional Minkowski space via a kind of Weierstrass representation [1]. Barbashov et al. studied the nonlinear differential equations describing in differential geometry the minimal surfaces in the pseudo-Euclidean space [2]. Kong et al. studied the equation of the relativistic string moving and the equation for the time-like extremal surfaces in the Minkowski space \(R^{1+n}\) [3, 4]. Liu and Zhou also gave the classical solutions to the initial boundary problem of time-like extremal surface [5, 6]. The time-like surfaces with vanishing mean curvature are constructed by [7, 8]. For the case of space-like extremal surfaces, we can see the classical papers of Calabi [9] and Cheng and Yau [10]. There are also important results for the purely space-like maximal surfaces [11, 12]. For the case of extremal surfaces of mixed type, we can also see the papers [12–15]. In addition, for the multidimensional cases, we refer to the papers by Lindblad [16], and Chae and Huh [17].
In the next section we will discuss the characteristic of extremal surface along a light-like line. We denote by \(y = \phi(x, t)\) the surface in Minkowski space \(R^{1+(1+1)}\). Many examples of space-like maximal surfaces containing singular curves have been constructed [18–20]. In particular, if one gives a generic regular light-like curve, then there exists a zero mean curvature surface which changes its causal type across this curve from a space-like maximal surface to a time-like minimal surface [12, 21–23]. This can be constructed by Weierstrass-type representation formula. However, if L is a light-like line, the construction fails since the isothermal coordinates break down along the light-like singular points. Locally, such surfaces are the graph of a function \(y = \phi(x, t)\) satisfying (1). We call this and its graph the zero mean curvature equation and zero mean curvature surface, respectively. Gu [12] and Klyachin [24] gave several fundamental results on zero mean curvature surfaces which might change type.
2 The properties and representations of extremal surface
2.1 The general formulas and analytic function
Theorem 1
The general expression of regular and dually regular time-like or space-like extremal surface in \(R^{1+(1+1)}\) is (12) or (13), respectively. If these two pieces can be matched regularly along the arc \(\rho=1\), \(a<\theta<b\), then the surface is analytic not only in the space-like part but also in the region \(a<\mu\leq\lambda<b\).
2.2 Global construction of extremal surfaces
Then we can construct the extension through the second borderline in a similar way.
3 Extremal surface along a light-like line
Proposition 3.1
If \(\mu> 0\) (\(\mu< 0\)), then the graph of \(y = \phi(x, t)\) is time-like (space-like) on both sides of L.
In particular, the graph might change type across L from space-like to time-like only if the constant μ vanishes. However, even in this case, the graph might not change type. We can normalize the constant μ to be −1, 0, 1. We can also get the general solutions to (24) and local existence of extremal surfaces with a light-like line.
Theorem 2
Theorem 3
For each positive number c, the formal power series solution \(\phi (x,t)\) uniquely determined by (32), (33), (34) and (35) gives a real analytic extremal surface on a neighborhood of \((x,t)=(0,0)\). In particular, there exists a non-trivial 1-parameter family of real analytic extremal surfaces, each of which changes type across a light-like line.
Proposition 3.2
In conclusion, we have finished the proof of Theorem 3 and given the local existence of extremal surfaces that change type beside a light-like line.
Notes
Declarations
Acknowledgements
The authors would like to thank Prof. Jianli Liu for his suggestions. The third author was partially supported by Doctoral Fund of Ministry of Education of People’s Republic of China (20133108120002).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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