## Extremal length and conformal geometry

December 6, 2010

There has been a couple of interesting talks recently here at Princeton. Somehow the term ‘extremal length’ came up in all of them. Due to my vast ignorance, I knew nothing about this before, but it sounded cool (and even somewhat systolic); hence I looked a little bit into that and would like to say a few words about it here.

One can find a rigorous exposition on extremal length in the book Quasiconformal mappings in the plane.

Let $\Omega$ be a simply connected Jordan domain in $\mathbb{C}$. $f: \Omega \rightarrow \mathbb{R}^+$ is a conformal factor on $\Omega$. Recall from my last post, $f$ is a Lebesgue measurable function inducing a metric on $\Omega$ where

$\mbox{Vol}_f(U) = \int_U f^2 dx dy$

and for any $\gamma: I \rightarrow \Omega$ ($I \subseteq \mathbb{R}$ is an interval) with $||\gamma'(t)|| = \bar{1}$, we have the length of $\gamma$:

$l_f(\gamma) = \int_I f dt$.

Call this metric $g_f$ on $\Omega$ and denote metric space $(\Omega, g_f)$.

Given any set $\Gamma$ of rectifiable curves in $U$ (possibly with endpoints on $\partial U$), each comes with a unit speed parametrization. Consider the “$f$-width” of the set $\Gamma$:

$\displaystyle w_f(\Gamma) = \inf_{\gamma \in \Gamma} l_f(\gamma)$.

Let $\mathcal{F}$ be the set of conformal factors $f$ with $L^2$ norm $1$ (i.e. having the total volume of $\Omega$ normalized to $1$).

Definition: The extremal length of $\Gamma$ is given by

$\mbox{EL}(\Gamma) = \displaystyle \sup_{f \in \mathcal{F}} w_f(\Gamma)^2$

Remark: In fact I think it would be more natural to just use $w_f(\Gamma)$ instead of $w_f(\Gamma)^2$ since it’s called a “length”…but since the standard notion is to sup over all $f$, not necessarily normalized, and having the $f$-width squared divide by the volume of $\Omega$, I can’t use conflicting notation. One should note that in our case it’s just the square of sup of width.

Definition:The metric $(\Omega, g_f)$ where this extremal is achieved is called an extremal metric for the family $\Gamma$.

The most important fact about extremal length (also what makes it an interesting quantity to study) is that it’s a conformal invariant:

Theorem: Given $h: \Omega' \rightarrow \Omega$ bi-holomorphic, then for any set of normalized curves $\Gamma$ in $\Omega$, we can define $\Gamma' = \{ h^{-1}\circ \gamma \ | \ \gamma \in \Gamma \}$ after renormalizing curves in $\Gamma'$ we have:

$\mbox{EL}(\Gamma) = \mbox{EL}(\Gamma')$

Sketch of a proof: (For simplicity we assume all curves in $\Gamma'$ are rectifiable, which is not always the case i.e. for bad maps $h$ the length might blow up when the curve approach $\partial \Omega'$ this case should be treated with more care)

This is indeed not hard to see, first we note that for any $f: \Omega \rightarrow \mathbb{R}^+$ we can define $f' : \Omega' \rightarrow \mathbb{R}^+$ by having

$f^\ast (z) = |h'(z)| (f \circ h) (z)$

It’s easy to see that $\mbox{Vol}_{f^\ast}(\Omega') = \mbox{Vol}_{f}(\Omega)$ (merely change of variables).

In the same way, $l_{f^\ast}(h^{-1}\circ \gamma) = l_f(\gamma)$ for any rectifiable curve.

Hence we have

$w_{f^\ast}(\Gamma') = w_f(\Gamma)$.

On the other hand, we know that $\varphi: f \mapsto f^\ast$ is a bijection from $\mathcal{F}_\Omega$ to $\mathcal{F}_{\Omega'}$, deducing

$\mbox{EL}(\Gamma) = \displaystyle ( \sup_{f \in \mathcal{F}} w_f(\Gamma))^2 = \displaystyle ( \sup_{f' \in \mathcal{F}'} w_{f'}(\Gamma'))^2 = \mbox{EL}(\Gamma')$

Establishes the claim.

One might wonder how on earth should this be applied, i.e. what kind of $\Gamma$ are useful to consider. Here we emphasis on the simple case where $\Omega$ is a rectangle (Of course I would first look at this case because of the unresolved issues from the last post :-P ):

Theorem: Let $R = (0,w) \times (0, 1/w)$, $\Gamma$ be the set of all curves starting at a point in the left edge $\{0\} \times [0, 1/w]$, ending on $\{1\} \times [0, 1/w]$ with finite length. Then $\mbox{EL}(\Gamma) = w^2$ and the Euclidean metric $f = \bar{1}$ is an extremal metric.

Sketch of the proof: It suffice to show that any metric $g_f$ with $\mbox{Vol}_f(R) = 1$ has at least one horizontal line segment $\gamma_y = [0,w] \times \{y\}$ with $l_f(\gamma_y) \leq w$. (Because if so, $w_f(\Gamma) \leq w$ and we know $w_{\bar{1}}(\Gamma) = w$ for the Euclidean length)

The average length of $\gamma_y$ over $y$ is

$w \int_0^{1/w} l_f(\gamma_y) dy$

$= w \int_0^{1/w} (\int_0^w f(t, y) dt) dy = w \int_R f$

By Cauchy-Schwartz this is less than $w (\int_R f^2)^{1/2} |R|^{1/2} = w$

Since the shortest curve cannot be longer than the average curve, we have $w_f(\Gamma) \leq w$.

Hence $\mbox{EL}(\Gamma) = \displaystyle \sup_{f \in \mathcal{F}}w_f(\Gamma)^2 = w^2$

Note it’s almost the same argument as in the proof of systolic inequality on the 2-torus.

Corollary: Rectangles with different eccentricity are not conformally equivalent (i.e. one cannot find a bi-homomorphic map between them sending each edge to an edge).

Remark: I was not aware of this a few days ago and somehow had the silly thought that there are conformal maps between any pair of rectangles while discussing with Guangbo >.< then tried to see what would those maps look like and was of course not able to do so. (there are obviously Riemann maps between the rectangles, but they don't send conners to conners, i.e. can't be extended to a conformal map on the closed rectangle).

An add-on: While I came across a paper of Odes Schramm, applying the techniques of extremal length, the following theorem seemed really cool.

Let $G = (V, E)$ be a finite planar graph with vertex set $V$ and edges $E \subseteq V^2$. For each vertex $v$ we assign a simply connected domain $D_v$.

Theorem: We can scale and translate each $D_v$ to $D'_v$ so that $\{ D_v \ | \ v \in V \}$ form a packing (i.e. are disjoint) and the contact graph of $D'_v$ is $G$. (i.e. $\overline{D'_{v_1}} \cap \overline{D'_{v_2}} \neq \phi$ iff $(v_1, v_2) \in E$.

Note: This is vastly stronger than producing a circle packing with prescribed structure.