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

Abstract

A. In this paper we investigate relations between solutions to the minimal surface equation in Euclidean 3-space E 3 , the zero mean curvature equation in Lorentz-Minkowski 3-space 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 noncommutativity between Wick rotations and isometries in the ambient space.2010 Mathematics Subject Classification. Primary 53A10; Secondary 58J72, 53B30.

Classification of zero mean curvature surfaces of separable type in Lorentz-Minkowski space

“…This maximal surface is a triply periodic surface and belongs to a family of triply periodic maximal surfaces which contains as particular examples, the H-type Schwarz surface and the D-type maximal surface [13]. Moreover, it is also the Scherk saddle tower under the motion of the Wick rotation of E 3 [2].…”

“…This surface is the right helicoid whose axis is the y-axis (spacelike axis) and known as the hyperbolic helicoid ( [15]) or the helicoid of the second kind ( [16]). Each helicoid is transformed into the other one by means of the Wick rotation [2].…”

“…These surfaces, the so-called rotational ZMC surfaces, are classified: see for example [16,18,23]. More recently, there is a great activity in the search of explicit examples of ZMC surfaces of mixed type where the transition between the spacelike and timelike regions occurs along degenerate curves: here we refer [2,9,10,11,12,13,15,22] without to be a complete list.…”

Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

“…On the other hand, there are several examples of ZMC-surfaces with degenerate light-like points (cf. [1,2,10,14]). Moreover, a local general existence theorem for maximal surfaces with degenerate light-like points is given in [21].…”

Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space

“…Klyachin [9] showed the following fact: This fact was generalized to a much wider class of surfaces, including real analytic constant mean curvature surfaces in R 3 1 , see [12,13]. Although there are properly embedded time-like ZMC-surfaces with an entire null line (see [3, Examples 2.2 and 2.3]), each of all examples of ZMC-surfaces with space-like points given in [1,2,6,10] containing an entire null line L has at least one cone-like singular point on L (see Fig. 2).…”