The Sym-Bobenko formula and constant mean curvature surfaces in Minkowski 3-space (Q1382855)

Recently, by means of the Weierstrass-type representation for harmonic maps from a Riemann surface into symmetric spaces discovered by Dorfmeister, Pedit and Wu, Dorfmeister and Haak have constructed constant mean curvature surfaces by applying the Sym-Bobenko formula to the loop-group-valued maps given by integrating the meromorphic potentials. On the other hand, Kenmotsu's work on the Weierstrass representation for immersions with prescribed mean curvature from a simply connected Riemann surface into Euclidean 3-space was generalized by \textit{K. Akutagawa} and \textit{S. Nishikawa} to Minkowski 3-space [Tôhoku Math. J., II. Ser. 42, 67-82 (1990; Zbl 0679.53002)]. Motivated by these results, the author of this paper considers two aspects. The first is to establish a natural correspondence between the following two spaces: the space of conformal spacelike immersions with constant mean curvature from a simply connected Riemann surface \(\Sigma\) into Minkowski 3-space, and that of nowhere anti-holomorphic harmonic maps from \(\Sigma\) into the Poincaré half plane, regarded as the Riemannian symmetric space \(SL(2, \mathbb{R})/SO(2)\). The second is to prove the Lorentzian version of the Sym-Bobenko formula and apply it to construct spacelike immersions with constant mean curvature into Minkowski 3-space. The main idea in this paper is to define an \(sl(2, \mathbb{R})\)-valued 1-form \(\Lambda^f\) on a Riemann surface \(\Sigma\) associated to a smooth map \(f:\Sigma\to SL(2, \mathbb{R})/SO(2)\) and show that the harmonicity of \(f\) is equivalent to the \(d\)-closedness of \(\Lambda^f\).