Weierstrass Representation for Timelike Minimal Surfaces in Minkowski 3-space

Using techniques of integrable systems, we study a Weierstraß representation formula for timelike surfaces with prescribed mean curvature in Minkowski 3-space. It is shown that timelike minimal surfaces are obtained by integrating a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in Minkowski 3-space. The relationship between timelike minimal surfaces and bosonic Nambu-Goto string worldsheets in spacetime is also discussed in the appendix. 1. Preliminaries In this section, we review some basics on the geometry of timelike surfaces in Minkowski 3-space. Let E2 be the semi-Euclidean 4-space with rectangular coordinates x0,x1,x2, x3 and the semi-Riemannian metric 〈 , 〉 of signature (−,−,+,+) given by the quadratic form ds = −dx0 − dx1 + dx2 + dx3. The semi-Euclidean 4-space E2 is identified with the linear space M2R of all 2× 2 real matrices via the correspondence (1) u = (x0, x1, x2, x3) ←→ ( x0 + x3 x1 + x2 −x1 + x2 x0 − x3 ) . This identification is an isometry, since 〈u,v〉 = 1 2 {tr(uv)− tr(u) tr(v)}, u,v ∈M2R. In particular, 〈u,u〉 = − detu. The standard basis e0, e1, e2, e3 for E2 is identified with with the matrices 1 = (