Uniformization of embedded surfaces

TL;DR

This paper explores the uniformization of a genus two surface embedded in the 3-sphere, providing explicit examples and computational verification using Maple to illustrate complex structures and conformal mappings.

Contribution

It offers explicit examples of uniformization for a specific embedded surface, supported by computational tools, filling a gap in explicit illustrations in the literature.

Findings

Explicit example of a genus two surface uniformization

Use of Maple for verification and numerical calculations

Insights into the rich theory behind surface uniformization

Abstract

Let X be a closed surface of genus two embedded in the 3-sphere. Then X inherits a metric and an orientation, which give an almost complex structure, which automatically integrates to a genuine complex structure, making X a Riemann surface. It follows that X is conformally isomorphic to a branched cover of the Riemann sphere, or to the quotient of the unit disc by the action of a Fuchsian group. The theorems behind these statements are important, well-known, and a century old. Nonetheless, we believe that the literature contains no examples where a significant fraction of the structure can be made explicit. This monograph is a partially successful attempt to provide such an example, starting with a particular surface X that has interesting geometry. The required theory is surprisingly rich, and is supported by a large body of Maple code, which is used for semi-formal verification of many proofs, as well as for numerical calculation.