Abstract
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.
Similar content being viewed by others
References
Akamine S, Causal characters of zero mean curvature surfaces of Riemann type in Lorentz-Minkowski 3-space, Kyushu J. Math. 71 (2017) 211–249
Akamine S, Behavior of the Gaussian curvature of timelike minimal surfaces with singularities, to appear in Hokkaido Math. J, arXiv:1701.00238
Born M and Infeld L, Foundations of the New Field Theory, Proc. R. Soc. London Ser. A. 144 852 (1934) 425–451
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
Clelland J N, Totally quasi-umbilic timelike surfaces in \({\mathbb{R}}^{1,2}\), Asian J. Math. 16 (2012) 189–208
Dey R, The Weierstrass–Enneper representation using hodographic coordinates on a minimal surface, Proc. Indian Acad. Sci. (Math. Sci.) 113(2) (2003) 189–193
Dey R and Singh R K, Born–Infeld solitons, maximal surfaces, Ramanujan’s identities, Arch. Math. 108(5) (2017) 527–538
Estudillo F J M and Romero A, Generalized maximal surfaces in Lorentz–Minkowski space \({\mathbb{L}}^3\), Math. Proc. Cambridge Phil. Soc. 111 (1992) 515–524
Fernández I, López F J and Souam R, The space of complete embedded maximal surfaces with isolated singularities in the \(3\)-dimensional Lorentz-Minkowski space, Math. Ann. 332 (2005) 605–643
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 \({\mathbb{L}}^3\) containing a light-like line, C.R. Acad. Sci. Paris. Ser. I 350 (2012) 975–978
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 \(3\)-space which change type across a light-like line, Osaka J. Math. 52 (2015) 285–297
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 \(3\)-space and \(2\)-dimensional fluid mechanics, Math. J. Okayama Univ. 57 (2015) 173–200
Gibbons G W and Ishibashi A, Topology and signature in braneworlds, Class. Quantum Gravit. 21 (2004) 2919–2935
Gu C, The extremal surfaces in the \(3\)-dimensional Minkowski space, Acta. Math. Sinica 1 (1985) 173–180
Kamien R D, Decomposition of the height function of Scherk’s first surface, Appl. Math. Lett. 14 (2001) 797–800
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
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
Klyachin V A, Zero mean curvature surfaces of mixed type in Minkowski space, Izv. Math. 67 (2003) 209–224
Kobayashi O, Maximal surfaces with cone-like singularities, J. Math. Soc. Japan 36 (1984) 609–617
Lee H, Extension of the duality between minimal surfaces and maximal surfaces, Geom. Dedicata 151 (2011) 373–386
López R, Time-like surfaces with constant mean curvature in Lorentz three-space, Tohoku Math. J. (2) 52(4) (2000) 515–532
Mallory M, Van Gorder R A and Vajravelu K, Several classes of exact solutions to the \(1+1\) Born-Infeld equation, Commun. Nonlinear Sci. Number. Simul. 19 (2014) 1669–1674
Umehara M and Yamada K, Maximal surfaces with singularities in Minkowski space, Hokkaido Math. J. 35 (2006) 13–40
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
Wick G C, Properties of Bethe–Salpeter wave functions, Phys. Rev. 96(4) (1954) 1124–1134
Acknowledgements
This study was initiated during the authors stay at the University of Granada in April 2017. They would like to thank Professor Rafael López for his invitation and hospitality. The first author was supported by Grant-in-Aid for JSPS Fellows Number 15J06677.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicating Editor: B V Rajarama Bhat
Rights and permissions
About this article
Cite this article
Akamine, S., Singh, R.K. Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born–Infeld equation. Proc Math Sci 129, 35 (2019). https://doi.org/10.1007/s12044-019-0479-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12044-019-0479-7