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Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born–Infeld equation


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.

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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.

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Correspondence to Shintaro Akamine.

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Communicating Editor: B V Rajarama Bhat

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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).

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  • Minimal surface
  • zero mean curvature surface
  • solution to the Born–Infeld equation
  • Wick rotation

2010 Mathematics Subject Classification

  • 53A10
  • 58J72
  • 53B30