Spacelike surfaces of constant mean curvature $\pm1$ in de Sitter $3$-space ${\mathbb S}^3_1(1)$

Abstract

It is shown that spacelike surfaces of constant mean curvature $\pm 1$ (abbreviated as CMC $\pm 1$) in de Sitter $3$-space ${\mathbb S}^3_1(1)$ can be constructed from holomorphic curves in ${\mathbb P}\mathrm{SL} (2;{\mathbb C})=\mathrm{SL} (2;{\mathbb C})/\{\pm\mathrm{id} \}$ via a Bryant type representation formula. This Bryant type representation formula is used to investigate an explicit one-to-one correspondence, the so-called {Lawson correspondence}, between spacelike CMC $\pm 1$ surfaces in de Sitter $3$-space ${\mathbb S}^3_1(1)$ and spacelike maximal surfaces in Lorentz $3$-space ${\mathbb E}^3_1$. The hyperbolic Gauss map of spacelike surfaces in ${\mathbb S}^3_1(1)$, which is a close analogue of the classical Gauss map, is considered. It is shown that the hyperbolic Gauss map plays an important role in the study of spacelike CMC $\pm 1$ surfaces in ${\mathbb S}^3_1(1)$. In particular, the relationship between the holomorphicity of the hyperbolic Gauss map and spacelike CMC $\pm 1$ surfaces in ${\mathbb S}^3_1(1)$ is studied.