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

TL;DR

This paper explores how solutions to minimal surface, zero mean curvature, and Born-Infeld equations are related through Wick rotations, revealing symmetries and transformations that connect real and imaginary solutions.

Contribution

It establishes a transformation theory linking solutions of these equations via Wick rotations and symmetries, providing new correspondences among solutions.

Findings

Wick rotations relate real and imaginary solutions through symmetries.

Transformation theory for zero mean curvature surfaces with lightlike lines.

New correspondences among solutions using non-commutativity of Wick rotations and isometries.

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 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 theory 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.