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

Abstract

We know that there exists the correspondence between one of the nonumbilical Riemannian surface with constant Gauss curvature K in M¯3(K¯)(K≠K¯), the nonumbilical spacelike or timelike surfaces with constant Gauss curvature K in M¯13(K¯)(K≠K¯), 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.