Spacelike Surfaces in [formula omitted] with null mean curvature vector and the nonlinear Riccati partial differential equation

Abstract

This paper introduces a complex Weierstrass representation for weakly conformal spacelike immersions in the Lorentz–Minkowski space L4 involving holomorphic or anti-holomorphic functions. Factoring through the light cone, we parametrize these immersions by three complex functions (μ,a,b). We prove the existence of a correspondence between those immersions for which a is holomorphic and b satisfies a Riccati partial differential equation, with conformal immersions in the hyperbolic space H3 with lightlike mean curvature vector. As a principal application of this correspondence, we obtain a powerful tool to construct new, explicit solutions of nonlinear Riccati partial differential equation and their explicit, associated surfaces. We also use our complex representation to classify the totally umbilic, weakly conformal immersions in L4, when a and b are both anti-holomorphic functions.