Fuchsian DPW potentials for Lawson surfaces

Abstract

The Lawson surfaces ${\xi }_{1,g}$ of genus $g$ are constructed by rotating and reflecting the Plateau solution $f_t$ with respect to a particular geodesic $4$-gon ${\Gamma }_t$ along its boundary, where $t=\frac{1}{2g+2}$ is an angle of ${\Gamma }_t$. In this paper we combine the existence and regularity of the Plateau solution $f_t$ in $t\in (0,\frac{1}{4})$ with topological information about the moduli space of Fuchsian systems on the 4-puncture sphere to obtain existence of a Fuchsian DPW potential ${\eta }_t$ for every $f_t$ with $t\in (0,\frac{1}{4}]$. Moreover, the coefficients of ${\eta }_t$ are shown to depend real analytically on $t$. This implies that the Taylor approximation of the DPW potential ${\eta }_t$ and of the area obtained at $t=0$ found in \cite{HHT2} determines these quantities for all ${\xi }_{1,g}$. In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces ${\xi }_{1,g}$.

21 pages