Paper detail

Uniformization of embedded surfaces

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.