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

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