Kenmotsu type representation formula for spacelike surfaces in the de Sitter 3-space (Q5933603)

Kenmotsu proved that the surfaces in space can be represented by means of their mean curvature and their Gauss map. Similar results have been obtained for surfaces in the hyperbolic 3-space, the surfaces in the 3-sphere, the spacelike surfaces in the Minkowski 3-space and the maximal surfaces in the anti-de Sitter 3-space (see references in this paper under review). This paper gives such a result for the spacelike surfaces in the de Sitter 3-space \(\mathbb{S}^3_1= SL(2,\mathbb{C})/SU(1,1)= \{ge_3g^*; g\in SL(2,\mathbb{C})\}\) with \(e_3=\text{diag}\{1,-1\}\), i.e., the semi-sphere of radius 1 in the Minkowski 4-space.NEWLINENEWLINENEWLINELet \(M\) be a Riemann surface, and \(f:M\to \mathbb{S}_1^3\) a conformal immersion whose image is a spacelike surface. Then, one can choose a so-called adapted framing \({\mathcal E}=({\mathcal E}_{ij}): M\to SL(2,\mathbb{C})\) of \(f\) locally, which is unique up to a \(U(1)\) factor on the right, where \(U(1)\) consists of the matrices of the form \(\text{diag} \{e^{i\theta}, e^{-i\theta}\}\) with \(\theta\in \mathbb{R}\). Thus, \(G={\mathcal E}_{21}/ \overline{{\mathcal E}_{22}}: M\to \mathbb{C}\cup \{\infty\}\) is globally defined (corresponds to the ``rotational part'' of \({\mathcal E})\) and is called the normal Gauss map of the surface. The mean curvature \(H\) and the normal Gauss map \(G\) satisfy NEWLINE\[NEWLINE\bigl(1-|G|^2\bigr) \biggl[1+ |G|^2+ \bigl(1-|G|^2 \bigr)H \biggr] G_{z\overline z}+2\biggl [|G|^2 +\bigl(1-|G|^2 \bigr)H \biggr]\overline GG_z G_{\overline z}=\bigl(1-|G|^2\bigr)^2 G_{\overline z}H_z.NEWLINE\]NEWLINE Moreover, the authors give the following converse. Let \(M\) be a simply connected Riemann surface with a reference point \(z_0\in M\) and an isothermal coordinate \(z,H:M\to \mathbb{R}\) a smooth function, and \(G:M\to \{w\in\mathbb{C}; |w|<1\}\) a non-holomorphic smooth function satisfying the above equation. If the 1-form NEWLINE\[NEWLINE\omega= {2(\overline G)_z\over \biggl[1+ |G|^2+ \bigl(1-|G|^2 \bigr)H\biggr] \bigl(1-|G|^2 \bigr)}dzNEWLINE\]NEWLINE is smooth on \(M\), then there exists a unique map \({\mathcal S}: M\to SL(2, \mathbb{C})\) such that \({\mathcal S}(z_0)=I\) and NEWLINE\[NEWLINE{\mathcal S}^{-1}d{\mathcal S}= {1 \over 2}(\beta+ \beta^*) e_3+{1\over 4}[e_3, \beta+\beta^*] e_3\text{ with } \beta= \left(\begin{matrix} G & 1\\ G^2 & G\end{matrix} \right)\omega,NEWLINE\]NEWLINE and \(f={\mathcal S} e_3{\mathcal S}^*:M\to \mathbb{S}^3_1\) is a conformal immersion outside the zero set of \(\omega\) with mean curvature \(H\) and normal Gauss map \(G\).