Fuchsian DPW potentials for Lawson surfaces

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

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