Extremal length

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In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves \Gamma is a conformal invariant of \Gamma. More specifically, suppose that D is an open set in the complex plane and \Gamma is a collection of paths in D and f:D\to D' is a conformal mapping. Then the extremal length of \Gamma is equal to the extremal length of the image of \Gamma under  f. For this reason, the extremal length is a useful tool in the study of conformal mappings. Extremal length can also be useful in dimensions greater than two, but the following deals primarily with the two dimensional setting.

Definition of extremal length[edit]

To define extremal length, we need to first introduce several related quantities. Let D be an open set in the complex plane. Suppose that \Gamma is a collection of rectifiable curves in D. If \rho:D\to [0,\infty] is Borel-measurable, then for any rectifiable curve \gamma we let

L_\rho(\gamma):=\int_\gamma \rho\,|dz|

denote the \rho-length of \gamma, where |dz| denotes the Euclidean element of length. (It is possible that L_\rho(\gamma)=\infty.) What does this really mean? If \gamma:I\to D is parameterized in some interval I, then \int_\gamma \rho\,|dz| is the integral of the Borel-measurable function \rho(\gamma(t)) with respect to the Borel measure on I for which the measure of every subinterval J\subset I is the length of the restriction of \gamma to J. In other words, it is the Lebesgue-Stieltjes integral \int_I \rho(\gamma(t))\,d{\mathrm{length}}_\gamma(t), where {\mathrm{length}}_\gamma(t) is the length of the restriction of \gamma to \{s\in I:s\le t\}. Also set

L_\rho(\Gamma):=\inf_{\gamma\in\Gamma}L_\rho(\gamma).

The area of \rho is defined as

A(\rho):=\int_D \rho^2\,dx\,dy,

and the extremal length of \Gamma is

EL(\Gamma):= \sup_\rho \frac{L_\rho(\Gamma)^2}{A(\rho)}\,,

where the supremum is over all Borel-measureable \rho:D\to[0,\infty] with 0<A(\rho)<\infty. If \Gamma contains some non-rectifiable curves and \Gamma_0 denotes the set of rectifiable curves in \Gamma, then EL(\Gamma) is defined to be EL(\Gamma_0).

The term modulus of \Gamma refers to 1/EL(\Gamma).

The extremal distance in D between two sets in \overline D is the extremal length of the collection of curves in D with one endpoint in one set and the other endpoint in the other set.

Examples[edit]

In this section the extremal length is calculated in several examples. The first three of these examples are actually useful in applications of extremal length.

Extremal distance in rectangle[edit]

Fix some positive numbers w,h>0, and let R be the rectangle R=(0,w)\times(0,h). Let \Gamma be the set of all finite length curves \gamma:(0,1)\to R that cross the rectangle left to right, in the sense that \lim_{t\to 0}\gamma(t) is on the left edge \{0\}\times[0,h] of the rectangle, and \lim_{t\to 1}\gamma(t) is on the right edge \{1\}\times[0,h]. (The limits necessarily exist, because we are assuming that \gamma has finite length.) We will now prove that in this case

EL(\Gamma)=w/h

First, we may take \rho=1 on R. This \rho gives A(\rho)=w\,h and L_\rho(\Gamma)=w. The definition of EL(\Gamma) as a supremum then gives EL(\Gamma)\ge w/h.

The opposite inequality is not quite so easy. Consider an arbitrary Borel-measurable \rho:R\to[0,\infty] such that \ell:=L_\rho(\Gamma)>0. For y\in(0,h), let \gamma_y(t)=i\,y+w\,t (where we are identifying \R^2 with the complex plane). Then \gamma_y\in\Gamma, and hence \ell\le L_\rho(\gamma_y). The latter inequality may be written as

 \ell\le \int_0^1 \rho(i\,y+w\,t)\,w\,dt .

Integrating this inequality over y\in(0,h) implies

 h\,\ell\le \int_0^h\int_0^1\rho(i\,y+w\,t)\,w\,dt\,dy.

Now a change of variable x=w\,t and an application of the Cauchy-Schwarz inequality give

 h\,\ell \le \int_0^h\int_0^w\rho(x+i\,y)\,dx\,dy \le \Bigl(\int_R \rho^2\,dx\,dy\int_R\,dx\,dy\Bigr)^{1/2} = \bigl(w\,h\,A(\rho)\bigr)^{1/2}. This gives \ell^2/A(\rho)\le w/h.

Therefore, EL(\Gamma)\le w/h, as required.

As the proof shows, the extremal length of \Gamma is the same as the extremal length of the much smaller collection of curves \{\gamma_y:y\in(0,h)\}.

It should be pointed out that the extremal length of the family of curves \Gamma\,' that connect the bottom edge of  R to the top edge of R satisfies EL(\Gamma\,')=h/w, by the same argument. Therefore, EL(\Gamma)\,EL(\Gamma\,')=1. It is natural to refer to this as a duality property of extremal length, and a similar duality property occurs in the context of the next subsection. Observe that obtaining a lower bound on EL(\Gamma) is generally easier than obtaining an upper bound, since the lower bound involves choosing a reasonably good \rho and estimating L_\rho(\Gamma)^2/A(\rho), while the upper bound involves proving a statement about all possible \rho. For this reason, duality is often useful when it can be established: when we know that EL(\Gamma)\,EL(\Gamma\,')=1, a lower bound on EL(\Gamma\,') translates to an upper bound on EL(\Gamma).

Extremal distance in annulus[edit]

Let r_1 and r_2 be two radii satisfying  0<r_1<r_2<\infty. Let A be the annulus A:=\{z\in\mathbb C:r_1<|z|<r_2\} and let C_1 and C_2 be the two boundary components of A: C_1:=\{z:|z|=r_1\} and C_2:=\{z:|z|=r_2\}. Consider the extremal distance in A between C_1 and C_2; which is the extremal length of the collection \Gamma of curves \gamma\subset A connecting C_1 and C_2.

To obtain a lower bound on EL(\Gamma), we take \rho(z)=1/|z|. Then for \gamma\in\Gamma oriented from C_1 to C_2

\int_\gamma |z|^{-1}\,ds \ge \int_\gamma |z|^{-1}\,d|z| = \int_\gamma d\log |z|=\log(r_2/r_1).

On the other hand,

A(\rho)=\int_A |z|^{-2}\,dx\,dy= \int_{0}^{2\pi}\int_{r_1}^{r_2} r^{-2}\,r\,dr\,d\theta = 2\,\pi \,\log(r_2/r_1).

We conclude that

EL(\Gamma)\ge \frac{\log(r_2/r_1)}{2\pi}.

We now see that this inequality is really an equality by employing an argument similar to the one given above for the rectangle. Consider an arbitrary Borel-measurable \rho such that \ell:=L_\rho(\Gamma)>0. For \theta\in[0,2\,\pi) let \gamma_\theta:(r_1,r_2)\to A denote the curve \gamma_\theta(r)=e^{i\theta}r. Then

\ell\le\int_{\gamma_\theta}\rho\,ds =\int_{r_1}^{r_2}\rho(e^{i\theta}r)\,dr.

We integrate over \theta and apply the Cauchy-Schwarz inequality, to obtain:

2\,\pi\,\ell \le \int_A \rho\,dr\,d\theta \le \Bigl(\int_A \rho^2\,r\,dr\,d\theta \Bigr)^{1/2}\Bigl(\int_0^{2\pi}\int_{r_1}^{r_2} \frac 1 r\,dr\,d\theta\Bigr)^{1/2}.

Squaring gives

4\,\pi^2\,\ell^2\le A(\rho)\cdot\,2\,\pi\,\log(r_2/r_1).

This implies the upper bound EL(\Gamma)\le (2\,\pi)^{-1}\,\log(r_2/r_1). When combined with the lower bound, this yields the exact value of the extremal length:

EL(\Gamma)=\frac{\log(r_2/r_1)}{2\pi}.

Extremal length around an annulus[edit]

Let r_1,r_2,C_1,C_2,\Gamma and A be as above, but now let \Gamma^* be the collection of all curves that wind once around the annulus, separating C_1 from C_2. Using the above methods, it is not hard to show that

EL(\Gamma^*)=\frac{2\pi}{\log(r_2/r_1)}=EL(\Gamma)^{-1}.

This illustrates another instance of extremal length duality.

Extremal length of topologically essential paths in projective plane[edit]

In the above examples, the extremal \rho which maximized the ratio L_\rho(\Gamma)^2/A(\rho) and gave the extremal length corresponded to a flat metric. In other words, when the Euclidean Riemannian metric of the corresponding planar domain is scaled by \rho, the resulting metric is flat. In the case of the rectangle, this was just the original metric, but for the annulus, the extremal metric identified is the metric of a cylinder. We now discuss an example where an extremal metric is not flat. The projective plane with the spherical metric is obtained by identifying antipodal points on the unit sphere in \R^3 with its Riemannian spherical metric. In other words, this is the quotient of the sphere by the map x\mapsto -x. Let \Gamma denote the set of closed curves in this projective plane that are not null-homotopic. (Each curve in \Gamma is obtained by projecting a curve on the sphere from a point to its antipode.) Then the spherical metric is extremal for this curve family.[1] (The definition of extremal length readily extends to Riemannian surfaces.) Thus, the extremal length is \pi^2/(2\,\pi)=\pi/2.

Extremal length of paths containing a point[edit]

If \Gamma is any collection of paths all of which have positive diameter and containing a point z_0, then EL(\Gamma)=\infty. This follows, for example, by taking

\rho(z):= \begin{cases}(-|z-z_0|\,\log |z-z_0|)^{-1} & |z-z_0|<1/2,\\
0 & |z-z_0|\ge 1/2,\end{cases}

which satisfies A(\rho)<\infty and L_\rho(\gamma)=\infty for every rectifiable \gamma\in\Gamma.

Elementary properties of extremal length[edit]

The extremal length satisfies a few simple monotonicity properties. First, it is clear that if \Gamma_1\subset\Gamma_2, then EL(\Gamma_1)\ge EL(\Gamma_2). Moreover, the same conclusion holds if every curve \gamma_1\in\Gamma_1 contains a curve \gamma_2\in \Gamma_2 as a subcurve (that is, \gamma_2 is the restriction of \gamma_1 to a subinterval of its domain). Another sometimes useful inequality is

EL(\Gamma_1\cup\Gamma_2)\ge \bigl(EL(\Gamma_1)^{-1}+EL(\Gamma_2)^{-1}\bigr)^{-1}.

This is clear if EL(\Gamma_1)=0 or if EL(\Gamma_2)=0, in which case the right hand side is interpreted as 0. So suppose that this is not the case and with no loss of generality assume that the curves in \Gamma_1\cup\Gamma_2 are all rectifiable. Let \rho_1,\rho_2 satisfy L_{\rho_j}(\Gamma_j)\ge 1 for j=1,2. Set \rho=\max\{\rho_1,\rho_2\}. Then L_\rho(\Gamma_1\cup\Gamma_2)\ge 1 and A(\rho)=\int\rho^2\,dx\,dy\le\int(\rho_1^2+\rho_2^2)\,dx\,dy=A(\rho_1)+A(\rho_2), which proves the inequality.

Conformal invariance of extremal length[edit]

Let f:D\to D^* be a conformal homeomorphism (a bijective holomorphic map) between planar domains. Suppose that \Gamma is a collection of curves in D, and let \Gamma^*:=\{f\circ \gamma:\gamma\in\Gamma\} denote the image curves under f. Then EL(\Gamma)=EL(\Gamma^*). This conformal invariance statement is the primary reason why the concept of extremal length is useful.

Here is a proof of conformal invariance. Let \Gamma_0 denote the set of curves \gamma\in\Gamma such that f\circ \gamma is rectifiable, and let \Gamma_0^*=\{f\circ\gamma:\gamma\in\Gamma_0\}, which is the set of rectifiable curves in \Gamma^*. Suppose that \rho^*:D^*\to[0,\infty] is Borel-measurable. Define

\rho(z)=|f\,'(z)|\,\rho^*\bigl(f(z)\bigr).

A change of variables w=f(z) gives

A(\rho)=\int_D \rho(z)^2\,dz\,d\bar z=\int_D \rho^*(f(z))^2\,|f\,'(z)|^2\,dz\,d\bar z = \int_{D^*} \rho^*(w)^2\,dw\,d\bar w=A(\rho^*).

Now suppose that \gamma\in \Gamma_0 is rectifiable, and set \gamma^*:=f\circ\gamma. Formally, we may use a change of variables again:

L_\rho(\gamma)=\int_\gamma \rho^*\bigl(f(z)\bigr)\,|f\,'(z)|\,|dz| = \int_{\gamma^*} \rho(w)\,|dw|=L_{\rho^*}(\gamma^*).

To justify this formal calculation, suppose that \gamma is defined in some interval I, let \ell(t) denote the length of the restriction of \gamma to I\cap(-\infty,t], and let \ell^*(t) be similarly defined with \gamma^* in place of \gamma. Then it is easy to see that d\ell^*(t)=|f\,'(\gamma(t))|\,d\ell(t), and this implies L_\rho(\gamma)=L_{\rho^*}(\gamma^*), as required. The above equalities give,

EL(\Gamma_0)\ge EL(\Gamma_0^*)=EL(\Gamma^*).

If we knew that each curve in \Gamma and \Gamma^* was rectifiable, this would prove EL(\Gamma)=EL(\Gamma^*) since we may also apply the above with f replaced by its inverse and \Gamma interchanged with \Gamma^*. It remains to handle the non-rectifiable curves.

Now let \hat\Gamma denote the set of rectifiable curves \gamma\in\Gamma such that f\circ\gamma is non-rectifiable. We claim that EL(\hat\Gamma)=\infty. Indeed, take \rho(z)=|f\,'(z)|\,h(|f(z)|), where h(r)=\bigl(r\,\log (r+2)\bigr)^{-1}. Then a change of variable as above gives

A(\rho)= \int_{D^*} h(|w|)^2\,dw\,d\bar w \le \int_0^{2\pi}\int_0^\infty (r\,\log (r+2))^{-2} \,r\,dr\,d\theta<\infty.

For \gamma\in\hat\Gamma and r\in(0,\infty) such that f\circ \gamma is contained in \{z:|z|<r\}, we have

L_\rho(\gamma)\ge\inf\{h(s):s\in[0,r]\}\,\mathrm{length}(f\circ\gamma)=\infty.[dubious ]

On the other hand, suppose that \gamma\in\hat\Gamma is such that f\circ\gamma is unbounded. Set H(t):=\int_0^t h(s)\,ds. Then L_\rho(\gamma) is at least the length of the curve t\mapsto H(|f\circ \gamma(t)|) (from an interval in \R to \R). Since \lim_{t\to\infty}H(t)=\infty, it follows that L_\rho(\gamma)=\infty. Thus, indeed, EL(\hat\Gamma)=\infty.

Using the results of the previous section, we have

EL(\Gamma)=EL(\Gamma_0\cup\hat\Gamma)\ge EL(\Gamma_0).

We have already seen that EL(\Gamma_0)\ge EL(\Gamma^*). Thus, EL(\Gamma)\ge EL(\Gamma^*). The reverse inequality holds by symmetry, and conformal invariance is therefore established.

Some applications of extremal length[edit]

By the calculation of the extremal distance in an annulus and the conformal invariance it follows that the annulus \{z:r<|z|<R\} (where 0\le r<R\le\infty) is not conformally homeomorphic to the annulus \{w:r^*<|w|<R^*\} if \frac Rr\ne \frac{R^*}{r^*}.

Extremal length in higher dimensions[edit]

The notion of extremal length adapts to the study of various problems in dimensions 3 and higher, especially in relation to quasiconformal mappings.

Discrete extremal length[edit]

Suppose that G=(V,E) is some graph and \Gamma is a collection of paths in G. There are two variants of extremal length in this setting. To define the edge extremal length, originally introduced by R. J. Duffin,[2] consider a function \rho:E\to[0,\infty). The \rho-length of a path is defined as the sum of \rho(e) over all edges in the path, counted with multiplicity. The "area" A(\rho) is defined as \sum_{e\in E}\rho(e)^2. The extremal length of \Gamma is then defined as before. If G is interpreted as a resistor network, where each edge has unit resistance, then the effective resistance between two sets of veritces is precisely the edge extremal length of the collection of paths with one endpoint in one set and the other endpoint in the other set. Thus, discrete extremal length is useful for estimates in discrete potential theory.

Another notion of discrete extremal length that is appropriate in other contexts is vertex extremal length, where \rho:V\to[0,\infty), the area is A(\rho):=\sum_{v\in V}\rho(v)^2, and the length of a path is the sum of \rho(v) over the vertices visited by the path, with multiplicity.

Notes[edit]

  1. ^ Ahlfors (1973)
  2. ^ Duffin 1962

References[edit]

  • Ahlfors, Lars V. (1973), Conformal invariants: topics in geometric function theory, New York: McGraw-Hill Book Co., MR 0357743 
  • Duffin, R. J. (1962), "The extremal length of a network", Journal of Mathematical Analysis and Applications 5 (2): 200–215, doi:10.1016/S0022-247X(62)80004-3 
  • Lehto, O.; Virtanen, K. I. (1973), Quasiconformal mappings in the plane (2nd ed.), Berlin, New York: Springer-Verlag