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.

Abstract. A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann type according to their causal characters, and as a corollary, we prove that if a zero mean curvature surface of Riemann type has exactly two causal characters, then the lightlike part of the surface is a part of a straight line.

We prove that the sign of the Gaussian curvature of any timelike minimal surface in the 3-dimensional Lorentz-Minkowski space is determined by the degeneracy and the orientations of the two null curves that generate the surface. We also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface which admits some kind of singular points.

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