Fuchsian DPW potentials for Lawson surfaces

The Lawson surface ξ _1,g of genus g is constructed by rotating and reflecting the Plateau solution f_t with respect to a particular geodesic 4-gon Γ _t across its boundary, where (t= 12g+2,π2,t,π2) are the angles of Γ _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 ∼ 0 . In this paper we combine the existence and regularity of the Plateau solution f_t in t ∈ (0, 14) 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 η _t for every f_t with t∈ (0, 14] . Moreover, the coefficients of η _t are shown to depend real analytically on t . This implies that the Taylor expansions of the DPW potential η _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 ξ _1,g . In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces ξ _1,g .