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Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born-Infeld equation. (English) Zbl 1417.53010

Summary: In this paper, we investigate relations between solutions to the minimal surface equation in Euclidean 3-space \(\mathbb{E}^3\), the zero mean curvature equation in the Lorentz-Minkowski 3-space \(\mathbb{L}^3\) 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 method 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 non-commutativity between Wick rotations and isometries in the ambient space.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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References:

[1] Akamine S, Causal characters of zero mean curvature surfaces of Riemann type in Lorentz-Minkowski 3-space, Kyushu J. Math.71 (2017) 211-249 · Zbl 1409.53010
[2] Akamine S, Behavior of the Gaussian curvature of timelike minimal surfaces with singularities, to appear in Hokkaido Math. J, arXiv:1701.00238 · Zbl 1429.53022
[3] Born M and Infeld L, Foundations of the New Field Theory, Proc. R. Soc. London Ser. A. 144852 (1934) 425-451 · Zbl 0008.42203
[4] Calabi E, Examples of Bernstein problems for some nonlinear equations, in Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, CA, 1968), Amer. Math. Soc., Providence, RI (1970) pp. 223-230
[5] Clelland J N, Totally quasi-umbilic timelike surfaces in \[{\mathbb{R}}^{1,2}\] R1,2, Asian J. Math.16 (2012) 189-208 · Zbl 1245.53018
[6] Dey R, The Weierstrass-Enneper representation using hodographic coordinates on a minimal surface, Proc. Indian Acad. Sci. (Math. Sci.)113(2) (2003) 189-193 · Zbl 1051.53004
[7] Dey R and Singh R K, Born-Infeld solitons, maximal surfaces, Ramanujan’s identities, Arch. Math.108(5) (2017) 527-538 · Zbl 1366.53007
[8] Estudillo F J M and Romero A, Generalized maximal surfaces in Lorentz-Minkowski space \[{\mathbb{L}}^3\] L3, Math. Proc. Cambridge Phil. Soc.111 (1992) 515-524 · Zbl 0824.53061
[9] Fernández I, López F J and Souam R, The space of complete embedded maximal surfaces with isolated singularities in the \[33\]-dimensional Lorentz-Minkowski space, Math. Ann.332 (2005) 605-643 · Zbl 1081.53052
[10] Fujimori S, Kim Y W, Koh S-E, Rossman W, Shin H, Takahashi H, Umehara M, Yamada K and Yang S-D, Zero mean curvature surfaces in \[{\mathbb{L}}^3\] L3 containing a light-like line, C.R. Acad. Sci. Paris. Ser. I350 (2012) 975-978 · Zbl 1257.53090
[11] Fujimori S, Kim Y W, Koh S-E, Rossman W, Shin H, Umehara M, Yamada K and Yang S-D, Zero mean curvature surfaces in Lorenz-Minkowski \[33\]-space which change type across a light-like line, Osaka J. Math.52 (2015) 285-297 · Zbl 1319.53008
[12] Fujimori S, Kim Y W, Koh S E, Rossman W, Shin H, Umehara M, Yamada K and Yang S-D, Zero mean curvature surfaces in Lorentz-Minkowski \[33\]-space and \[22\]-dimensional fluid mechanics, Math. J. Okayama Univ.57 (2015) 173-200 · Zbl 1320.53017
[13] Gibbons G W and Ishibashi A, Topology and signature in braneworlds, Class. Quantum Gravit.21 (2004) 2919-2935 · Zbl 1064.83069
[14] Gu C, The extremal surfaces in the \[33\]-dimensional Minkowski space, Acta. Math. Sinica1 (1985) 173-180 · Zbl 0595.49027
[15] Kamien R D, Decomposition of the height function of Scherk’s first surface, Appl. Math. Lett.14 (2001) 797-800 · Zbl 1033.53006
[16] Kim Y W, Koh S-E, Shin H and Yang S-D, Space-like maximal surfaces, time-like minimal surfaces, and Björling representation formulae, J. Korean Math. Soc.48 (2011) 1083-1100 · Zbl 1250.53016
[17] Kim Y W and Yang S-D, Prescribing singularities of maximal surfaces via a singular Björling representation formula, J. Geom. Phys.57 (2007) 2167-2177 · Zbl 1141.53012
[18] Klyachin V A, Zero mean curvature surfaces of mixed type in Minkowski space, Izv. Math.67 (2003) 209-224 · Zbl 1076.53015
[19] Kobayashi O, Maximal surfaces with cone-like singularities, J. Math. Soc. Japan36 (1984) 609-617 · Zbl 0548.53006
[20] Lee H, Extension of the duality between minimal surfaces and maximal surfaces, Geom. Dedicata151 (2011) 373-386 · Zbl 1211.53010
[21] López R, Time-like surfaces with constant mean curvature in Lorentz three-space, Tohoku Math. J. (2)52(4) (2000) 515-532 · Zbl 0981.53051
[22] Mallory M, Van Gorder R A and Vajravelu K, Several classes of exact solutions to the \[1+11+1\] Born-Infeld equation, Commun. Nonlinear Sci. Number. Simul.19 (2014) 1669-1674 · Zbl 1457.78004
[23] Umehara M and Yamada K, Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J.35 (2006) 13-40 · Zbl 1109.53016
[24] Umehara M and Yamada K, Surfaces with light-like points in Lorentz-Minkowski space with applications, in: Lorentzian Geometry and Related Topics, Springer Proc. Math Statics (2017) vol. 21, pp. 253-273 · Zbl 1402.53007
[25] Wick G C, Properties of Bethe-Salpeter wave functions, Phys. Rev.96(4) (1954) 1124-1134 · Zbl 0057.21202
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