Reflection principle for lightlike line segments on maximal surfaces
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Reflection principle for lightlike line segments on maximal surfaces Shintaro Akamine1 · Hiroki Fujino2 Received: 30 August 2020 / Accepted: 5 October 2020 © Springer Nature B.V. 2020
Abstract As in the case of minimal surfaces in the Euclidean 3-space, the reflection principle for maximal surfaces in the Lorentz-Minkowski 3-space asserts that if a maximal surface has a spacelike line segment L, the surface is invariant under the 180◦-rotation with respect to L. However, such a reflection property does not hold for lightlike line segments on the boundaries of maximal surfaces in general. In this paper, we show some kind of reflection principle for lightlike line segments on the boundaries of maximal surfaces when lightlike line segments are connecting shrinking singularities. As an application, we construct various examples of periodic maximal surfaces with lightlike lines from tessellations of ℝ2. Keywords Reflection principle · Maximal surface · Lightlike boundary problem · Harmonic mapping Mathematics Subject Classification 53A10 · 53B30 · 31A05 · 31A20
1 Introduction The classical Schwarz reflection principle for harmonic functions yields a symmetry principle for minimal surfaces in the 3-dimensional Euclidean space 𝔼3 : if a minimal surface in 𝔼3 has a straight line segment L on its boundary, then the surface can be extended across L and the extended surface is invariant under the 180◦-rotation with respect to L (see [6, p. 289], [24, p. 140] and [25, p. 54] for example). This principle is directly derived from The first author was partially supported by JSPS KAKENHI Grant Number 19K14527, 17H06466 and JSPS/FWF Bilateral Joint Project I3809-N32 “Geometric Shape Generation”, and the second author by JSPS KAKENHI Grant Number 19K21022 and 20K14306. * Shintaro Akamine akamine.shintaro@nihon‑u.ac.jp Hiroki Fujino [email protected]‑u.ac.jp 1
Department of Liberal Arts, College of Bioresource Sciences, Nihon University, 1866 Kameino, Fujisawa, Kanagawa 252‑0880, Japan
2
Institute for Advanced Research, Graduate School of Mathematics, Nagoya University, Chikusa‑ku, Nagoya 464‑8602, Japan
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Annals of Global Analysis and Geometry
the Schwarz reflection principle and the fact that each coordinate function of a conformal minimal immersion in 𝔼3 is harmonic. For the same reason, such a reflection principle for lines also holds for maximal surfaces, i.e., spacelike surfaces with vanishing mean curvature, in the three-dimensional Lorentz–Minkowski space 𝕃3 , when the straight line segment is spacelike, see Alías–Chaves–Mira [5, Theorem 3.10]. As a singular version of this reflection principle, a reflection principle inducing point symmetries for shrinking or conelike singularities was also shown in [10, 11, 21, 23] (see also [12, 15, 17, 22]). These regular and singular versions of reflection principles for maximal surfaces are highly depend on the conformal structures of surfaces. However, as another possibility of a line reflection principle, lightlike line segments can appear on th
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