Timelike surfaces in the de Sitter space $\mathbb S^3_1(1)\subset \mathbb R^4_1$

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 we obtain a representation formula. Real and complex special quadrics in $\mathbb CP^3$ are identified with the grassmannians of spacelike and timelike oriented 2-planes of $\mathbb R^4_1$, and the normal frame is written in terms of certain complex valued functions $x$ and $y$, which may be considered holomorphic functions as a special case. Then several results describing the analytic restrictions via solutions of certain PDE in complex variable, are shown. Finding solutions allows us to identify explicitly the representation of the associated surfaces. Moreover, using our technique we find a new kind of complex function which we call quasi-holomorphic and which satisfy a generalized version of the Cauchy-Riemann equations. Our technique allows the explicit construction of 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.