Fuchsian DPW potentials for Lawson surfaces

Abstract

The Lawson surface $\xi _{1,g}$ of genus g is constructed by rotating and reflecting the Plateau solution $f_t$ with respect to a particular geodesic 4-gon $\Gamma _t$ across its boundary, where $(t= \tfrac{1}{2g+2},\tfrac{\pi }{2},t,\tfrac{\pi }{2})$ are the angles of $\Gamma _t$. Recent progress in integrable surface theory allows for a more explicit construction of these surfaces and for better understanding of their geometric properties using so-called Fuchsian DPW potentials for $t \sim 0$. In this paper we combine the existence and regularity of the Plateau solution $f_t$ in $t \in (0, \tfrac{1}{4})$ with a detailed investigation of the moduli space of Fuchsian systems on the 4-punctured sphere to obtain the existence of a Fuchsian DPW potential $\eta _t$ for every $f_t$ with $t\in (0, \tfrac{1}{4}]$. Moreover, the coefficients of $\eta _t$ are shown to depend real analytically on t. This implies that the Taylor expansions of the DPW potential $\eta _t$ and of the area in Heller et al. (Complete families of embedded high genus CMC surfaces in the 3-sphere. arXiv:2108.10214) computed at $t=0$ already determine these quantities for all $\xi _{1,g}$. In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces $\xi _{1,g}$.