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

Proceedings of the Indian Academy of Sciences. Mathematical sciences

SHINTARO AKAMINE^1¹  RAHUL KUMAR SINGH^2² 

[1] Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8502, Japan^1;School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneshwar, Khurda, Odisha 752 050, India^2

关键词: Minimal surface;    zero mean curvature surface;    solution to the Born–Infeld equation;    Wick rotation;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 light like 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.

【 授权许可】

CC BY