Timelike surfaces in the de Sitter space $$\mathbb S_1^3(1)\subset {{\mathbb R}_{1}^{4}}$$

Abstract

This paper studies timelike minimal surfaces in the De Sitter space $$\mathbb S^3_1(1) \subset \mathbb R^4_1$$ via a complex variable. Using complex analysis and stereographic projection of lightlike vectors in $$\mathbb C \cup \{\infty \}$$ , we obtain a complex representation formula, together with some results about the existence of convenient isotropic coordinates. This allows us to construct timelike minimal surfaces in $$\mathbb S^3_1(1)$$ via local solutions of a certain PDE in a complex variable which arises when investigating our geometric conditions. Specifically, we find a new kind of complex functions which generalize the classes of holomorphic and anti-holomorphic functions, which we call quasi-holomorphic functions. We show that there is a correspondence between a timelike minimal surface in $$\mathbb S^3_1(1)$$ and a pair of quasi-holomorphic functions. In particular, when the two functions are holomorphic, we show that they are related by a Möbius transformation and then construct many families of minimal timelike surfaces in $$\mathbb S^3_1(1)$$ whose intrinsic Gauss map will also belong to the same class of surfaces. Several explicit examples are given.