Weierstrass Representations of Lorentzian Minimal Surfaces in $$\mathbb R^4_2$$ R 2 4
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Weierstrass Representations of Lorentzian Minimal Surfaces in R42 Ognian Kassabov and Velichka Milousheva Abstract. The minimal Lorentzian surfaces in R42 whose first normal space is two-dimensional and whose Gauss curvature K and normal curvature κ satisfy K 2 − κ 2 > 0 are called minimal Lorentzian surfaces of general type. These surfaces admit canonical parameters and with respect to such parameters are determined uniquely up to a motion in R42 by the curvatures K and κ satisfying a system of two natural PDEs. In the present paper we study minimal Lorentzian surfaces in R42 and find a Weierstrass representation with respect to isothermal parameters of any minimal surface with two-dimensional first normal space. We also obtain a Weierstrass representation with respect to canonical parameters of any minimal Lorentzian surface of general type and solve explicitly the system of natural PDEs expressing any solution to this system by means of four real functions of one variable. Mathematics Subject Classification. Primary 53B30; Secondary 53A10, 53A35. Keywords. Lorentzian surfaces, Weierstrass formulas, Canonical principal parameters, Minimal surfaces.
1. Introduction The study of minimal surfaces is one of the main topics in classical differential geometry. In the last years, great attention is paid to Lorentzian surfaces in pseudo-Euclidean spaces, since pseudo-Riemannian geometry has many important applications in Physics. Minimal Lorentzian surfaces in C21 have been classified by Chen [5]. Classification results for minimal Lorentzian surfaces in pseudo-Euclidean space Rm s with arbitrary dimension m and arbitrary index s are obtained in Ref. [6]. Minimal Lorentzian surfaces in the pseudoEuclidean 4-space R42 whose first normal space is two-dimensional and whose Gauss curvature K and normal curvature κ satisfy the inequality K 2 −κ 2 > 0 are studied in Ref. [1] under the name minimal Lorentzian surfaces of general 0123456789().: V,-vol
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type. This class of surfaces is characterized in terms of a pair of smooth functions satisfying a system of two natural partial differential equations. The approach to the study of minimal Lorentzian surfaces of general type in R42 is based on the introducing of special geometric parameters which are called canonical parameters. A representation of a minimal Lorentzian surface was given by Dussan and Magid in Ref. [8] where they solved the Bj¨orling problem for timelike surfaces in R42 constructing a special normal frame and a split-complex representation formula. The Bj¨ orling problem for timelike surfaces in the Lorentz–Minkowski spaces R31 and R41 is solved in Refs. [4,7], respectively. Spinor representation of Lorentzian surfaces in the pseudo-Euclidean 4-space with neutral metric is given in Ref. [3]. In Ref. [14], Patty gave a generalized Weierstrass representation of a minimal Lorentzian surface in R42 using spinors and Lorentz numbers (also known as para-complex, split-complex, double or hyperbolic numbers) thus extending
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