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Simultaneous Uniformization Theorem October 23, 2010

Posted by Sarah in Uncategorized.
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The other day, in the graduate student talks, Subhojoy was talking about the Simultaneous Uniformization Theorem. It was a nice treat because I used to be really into geometric topology (or at least as much as an undergrad with way too little background could.)

The big reveal is
QF(S) \simeq T(S) \times T(S)
but most of the talk, naturally, goes into defining what those letters mean.

The Riemann Mapping Theorem says that any simply connected set U \subset \mathbb{C} is conformally equivalent to the disc \mathbb{D}. Conformal maps are angle-preserving; in practice, they’re holomorphic functions with derivative everywhere nonzero. Conformal maps take round circles to round circles if the circles are small enough.

A Riemann Surface is a topological surface with conformal structure: a collection of charts to the complex plane such that the transition maps are conformal.

The first version of the Uniformization Theorem says that any simply-connected Riemann surface is conformally equivalent to the Riemann sphere (or complex plane or unit disc; these are all equivalent.)

The second, more general version of the Uniformization Theorem says that any Riemann surface of genus \ge 2 is conformally equivalent to \mathbb{H}^2/\Gamma where \mathbb{H}^2 is the hyperbolic plane and \Gamma is a discrete subgroup of PSL_2 \mathbb{R}.

To understand this better, we should observe more about the universal cover of a Riemann surface. This is, of course, simply connected. Its deck transformations are conformal automorphisms of the disc. But it can be proven that conformal automorphisms of the disc are precisely the Mobius transformations, or functions of the form

\frac{az + b}{cz+d}

This implies that the automorphism group of \mathbb{D} is PSL_2\mathbb{R}.

Now observe that there’s a model of the hyperbolic plane on the disc, by assigning the metric

\frac{4(dr^2 + r^2 d\theta^2)}{(1-r^2)^2}.

And, if you were to check, it would turn out that conformal transformations on the disc preserve this metric.

So it begins to make sense; Riemann surfaces are conformally equivalent to their universal covering space, modulo some group \Gamma of relations, a subgroup of the group of deck transformations of the universal cover.

\Gamma are called Fuchsian groups — these define which Riemann surface we’re talking about, up to a conformal transformation.

Now we can define Fuchsian space as

F(S) = \{ \rho: \pi_1(S) \to PSL_2\mathbb{R} | \rho \mathrm{discrete, injective} \}

It’s the set of maps from the fundamental group of a surface to PSL_2\mathbb{R}.

And we can define Teichmuller space as the space of marked conformal structures on the surface S.
This is less enormously huge than you might think, because we consider these up to an equivalence relation. If f: S \to X and g: S \to Y are conformal structures, and there exists a conformal map h: X \to Y such that h \circ f = g then we consider (f, X) \sim (g, Y) equivalent structures.

In fact, Teichmuller space is not that enormously huge: T(S) \simeq \mathbb{R}^{6g-6}. It turns out that Teichmuller space is completely determined by what happens to the boundary circles in a pair of pants decomposition of the surface.

Here’s a picture of a pair of pants (aka a three-punctured sphere):

One pair of pants:

Here’s a picture of a decomposition of a Riemann surface into pairs of pants:

A pair of pants decomposition of a Riemann surface:

(Here’s a nice article demonstrating the fact. It’s actually not as hard as it looks.)

Now we generalize to Quasi-Fuchsian spaces. For this, we’ll be working with hyperbolic 3-space instead of 2-space. The isometries of hyperbolic 3-space happen to be PSL_2 \mathbb{C}.
Instead of a Poincare Disc Model, we have a ball model; again, PSL_2 \mathbb{C} acts by Mobius transformations, functions of the form \frac{az+b}{cz+d}.

A quasiconformal function takes, infinitesimally, circles to ellipses. It’s like a conformal map, but with a certain amount of distortion. The Beltrami coefficient definds how much distortion:

\mu = \frac{f_{\bar{z}}}{f_z}

Quasi-Fuchsian space, QF(S), is the set of all quasiconformal deformations of a Fuchsian representation. In other words, this is the space of all representations to PSL_2 \mathbb{C} preserving topological circles on the boundary.

Now, the Simultaneous Uniformization Theorem can be stated:
the Quasi-Fuchsian space of a surface is isomorphic to the product of two Teichmuller spaces of the surface.

One application of this theorem is to hyperbolic 3-manifolds.
If M = \mathbb{H}^3/\Gamma is a hyperbolic 3-manifold, and if \Gamma \simeq S then $M \simeq S \times \mathbb{R}$.

In other words, we can think of a hyperbolic three-manifold as the real line, with a Riemann surface at each point — you can only sort of visualize this, as it’s not embeddable in 3-space.

The Simultaneous Uniformization Theorem implies that there is a hyperbolic metric on this 3-manifold for any choice of conformal structure at infinity.

This contrasts with the Mostow Rigidity Theorem, which states that a closed 3-manifold has at most one hyperbolic structure.

Together, these statements imply that any hyperbolic metric on S \times \mathbb{R} is determined uniquely by the choice of conformal structures at infinity.



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