This page is associated with the monograph
Uniformization of embedded surfaces
by
Neil Strickland,
and the related Maple code.
Let $X\subset S^3$ be a smoothly embedded closed surface of genus
$g\gt 1$. As we will explain, $X$ automatically has a very rich and
rigid geometric structure. Indeed, $X$ inherits a Riemannian metric
from $S^3$, and after specifying some conventions we also obtain a
well-defined orientation. Now consider a point $x\in X$, and let
$T_xX$ denote the corresponding tangent space. Let $J_x\colon T_xX\to
T_xX$ be the anticlockwise rotation through $\pi/2$ (which is
meaningful given the metric and orientation). This depends smoothly
on $x$ and satisfies $J_x^2=-1$, so it gives an almost complex
structure on $X$. It has been known since the early twentieth century
that any almost complex structure on a manifold of real dimension two
integrates to give a genuine complex structure. Thus, $X$ can be
regarded as a compact Riemann surface. It is known that any compact
Riemann surface can be regarded as a projective algebraic variety over
$\mathbb{C}$, and also as a branched cover of the Riemann sphere.
Alternatively, as we have assumed that the genus is larger than one,
the universal cover of $X$ is conformally equivalent to the open unit
disc $\Delta$. This means that $X$ is conformally equivalent to the
quotient $\Delta/\Pi$ for some Fuchsian group $\Pi$.
To the best of our knowledge, the literature contains no examples
where a significant fraction of this structure can be made explicit.
This project is a partially successful attempt to provide such
an example, involving the surface
\[ EX^* = \{x\in S^3\; | \;
(3x_3^2-2)x_4+\sqrt{2}(x_1^2-x_2^2)x_3=0
\}
\]