Uniformization of embedded surfaces

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 \[ EX^* = \{x\in S^3\; | \; (3x_3^2-2)x_4+\sqrt{2}(x_1^2-x_2^2)x_3=0 \} \]