Timelike surfaces of constant mean and constant Gaussian curvature in anti-de Sitter 3-space via loop groups

Abstract

The loop group description of surfaces whose Gauss map is Lorentz harmonic is known for timelike surfaces of constant mean curvature in the flat Lorentzian 3-space, and for surfaces of constant Gaussian curvature in the 3-sphere. We carry out the corresponding construction for timelike surfaces in anti-de Sitter 3-space $\mathbb{H}^3_1\cong SL(2,\mathbb{R})$: those of constant mean curvature, whose Gauss map is harmonic for the conformal structure of the first fundamental form, and those of constant Gaussian curvature $K>-1$, $K\neq0$, whose Gauss map is then an immersion and is harmonic for the second fundamental form. For both classes we prove the harmonicity characterization, recover the surfaces from extended frames by Sym-Bobenko-type formulas, and solve the geometric Cauchy problem by the generalized DPW method, with potentials written explicitly in terms of the data and no normalization of the Gauss map. We then show that the two constructions differ only by a normalization: after the substitution $λ=ζ^2$ and a constant gauge, a constant mean curvature extended frame $\hat F$ is an extended frame of its Gauss map, and the maps $\hat F|_{λ=e^{4θ}}\exp(θe_3)\hat F|_{λ=1}^{-1}$ and $e_3\hat F|_{λ=e^{4θ}}e_3\exp(θe_3)\hat F|_{λ=1}^{-1}$, with $e_3=\mathrm{diag}(-1,1)$, are timelike surfaces of constant Gaussian curvature $1/\sinh^2θ$ and $-1/\cosh^2θ$ wherever the constant mean curvature surface is not flat. The first is the classical parallel surface, the second its polar surface. Explicit families, with extended frames and surfaces in closed form, illustrate the constructions.