An appendix to the Weierstrass representation of a maximal spacelike surface in \(\mathbb{L}^3\)
The author provides some additional information on the Weierstrass pair \(\{g,\eta\}\) representing a maximal surface in Lorentz-Minkowski 3-space \(\mathbb L^3\). Let \(\mathbf F:M\to\mathbb L^3\) be a maximal immersion of an oriented and connected surface so that the unit (time-like) normal \(\mathbf N\) lies in the hyperbolic plane \(\mathbb H^2= \{(r,s,t): r^2+s^2-t^2= -1\), \(t\geq 1\}\) and the Riemann surface structure on \(M\) is compatible with the choice of \(\mathbf N\). Then to it corresponds a Weierstrass pair \(\{g,\eta\}\) so that \(\mathbf F\) is represented by \(\mathbf F=(1/2)\mathfrak R(\int(1+g^2,i (1-g^2),2g)\eta)\). The map \(g\) and 1-form \(\eta\) are globally defined and holomorphic and \(g=\Pi\circ\mathbf N,\) where \(\Pi(r,s,t)= (r+is)/(1+t)\), maps \(\mathbb H^2\) isometrically onto the unit disc \(\mathbb D\). The author proves that \(g\) is holomorphic with values in \(\mathbb D\) (i.e. \(|g|<1\)) and that \(\eta\) is a holomorphic 1-form that never vanishes. This is a refinement of a result of \textit{O. Kobayashi} [Tokyo J. Math. 6, 297--309 (1983; Zbl 0535.53052)], who showed only that for a maximal surface of the 1st kind \(g\) is a meromorphic function with \(|g|\neq 1\) and \(\eta\) is a holomorphic 1-form such that \(g^2\eta\) is holomorphic and does not vanish on the zeroes of \(\eta\).
scientific article; zbMATH DE number 5545416
scientific article