Generalized Weierstrass representations of surfaces with the constant Gauss curvature in pseudo-Riemannian three-dimensional space forms

We know that there exists the correspondence between one of the nonumbilical Riemannian surface with constant Gauss curvature K in (M) over bar (3) ((K) over bar) (K not equal (K) over bar), the nonumbilical spacelike or timelike surfaces with constant Gauss curvature K in (M) over bar (3)(1) ((K) over bar) (K not equal (K) over bar), and a solution of one of the following differential equations: the sine-Gordon, sinh-Gordon, sine-laplace, sinh-laplace, and cosh-Gordon equations. In this paper, we consider the initial value problems of these equations and give the generalized Weierstrass representations of these surfaces that depend only on the initial values of these equations. (c) 2007 American Institute of Physics.