Lawson's Surface Uniformization

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Description

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The Lawson surface is the only known minimal surface in $\mathbb S^3$ with genus $2$. We present a discrete uniformization of the corresponding Riemann surface as constructed in [1]. We present the surface in three different representations and show the corresponding universal coverings.

Embedding in $\mathbb R^3$

We choose a canonical fundamental domain and a base point that corresponds to north/south pole of the corresponding algebraic curve, see Figure [Embedded Surface]. The surface is a stereographic projection of the Lawson minimal surface in $\mathbb S^3$.

Algebraic Curve Representation

The branch points of the curve $\mu^2=\lambda^6-1$ are the $6$ roots of unity. On $\mathbb S^2$ we place them on the equator. The discrete curve is a doubly covered triangulated sphere branched at these points, see Figure [Triangulated Branched Cover].

Square Tiled Surface

The Lawson surface is conformally equivalent to the square tiled surface depicted in Figure [Square Tiled Representation]. We choose a fundamental domain and group presentation particularly suited for this representation. Namely we cut the surface such that the six quads are arranged around a common vertex. The branch points of the corresponding algebraic curve are the centers of the quads, see Figure [Universal Cover of Square Tiled Surface]. Figures [Embedded Surface Square Tiled Domain] and [Branched Cover Square Tiled Domain] show the analogous fundamental domains for the embedded surface and the algebraic curve representation.

XML Data

We give data for the map and the uniformizing groups as XML data. See files at [Embedded Surface], [Triangulated Branched Cover], and [Square Tiled Representation], as well as for [Embedded Surface Square Tiled Domain] and [Branched Cover Square Tiled Domain].

For details on the construction and mathematical underpinnings see also [2] and [3].

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