Minimal timelike surfaces in the Lorentz–Minkowski 3-space and their canonical parameters

Abstract

We study minimal timelike surfaces in \(\mathbb R^3_1\) using a special Weierstrass-type formula in terms of holomorphic functions defined in the algebra of the double (split-complex) numbers. We present a method of obtaining an equation of a minimal timelike surface in terms of canonical parameters, which play a role similar to the role of the natural parameters of curves in \(\mathbb R^3\). Having one holomorphic function that generates a minimal timelike surface, we find all holomorphic functions that generate the same surface. In this way we give a correspondence between a minimal timelike surface and a class of holomorphic functions. As an application, we prove that the Enneper surfaces are the only minimal timelike surfaces in \(\mathbb R^3_1\) with polynomial parametrization of degree 3 in isothermal parameters.