Fuchsian DPW potentials for Lawson surfaces

TL;DR

This paper establishes the existence of Fuchsian DPW potentials for Lawson surfaces, showing their coefficients depend analytically on a parameter, enabling a conformal parametrization algorithm for these surfaces.

Contribution

It combines Plateau solution regularity with Fuchsian system topology to prove the existence of DPW potentials for all Lawson surfaces in a specific family.

Findings

Existence of Fuchsian DPW potentials for all relevant Lawson surfaces.

Analytic dependence of potential coefficients on the parameter t.

Algorithm for conformally parametrizing 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 $t= \tfrac{1}{2g+2}$ is an angle of $\Gamma_t$. In this paper we combine the existence and regularity of the Plateau solution $f_t$ in $t \in (0, \tfrac{1}{4})$ with topological information about the moduli space of Fuchsian systems on the 4-puncture sphere to obtain existence of a Fuchsian DPW potential $\eta_t$ 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 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 $\xi_{1,g}$.