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

2017/11/01 by Shintaro Akamine, Akamine, Shintaro, Rahul Kumar Singh +1 · 2 citations
Mathematics · Physics and Astronomy · #53A10 (Primary) #53B30 (Secondary) #58J72 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1711.00299

openalex publication_date 2017/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we investigate relations between solutions to the minimal surface equation in Euclidean 3-space 𝔼³, the zero mean curvature equation in Lorentz-Minkowski 3-space \mathbbL³ 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.

Citations

• Behavior of the Gaussian curvature of timelike minimal surfaces with singularities
• Surfaces with light-like points in Lorentz-Minkowski 3-space with applications

Cited by

• Decompositions of Scherk-Type Zero Mean Curvature Surfaces
• Examples of entire zero-mean curvature graphs of mixed-type in Lorentz-Minkowski space via Konderak's formulas

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