Extremal length

To define extremal length, we need to first introduce several related quantities.

Let $D$ be an open set in the complex plane. Suppose that $\backslash Gamma$ is a

collection of rectifiable curves in $D$. If $\backslash rho:D\backslash to\; [0,\backslash infty]$

is Borel-measurable, then for any rectifiable curve $\backslash gamma$ we let

:$L\_\backslash rho(\backslash gamma):=\backslash int\_\backslash gamma\; \backslash rho\backslash ,|dz|$

denote the $\backslash rho$–length of $\backslash gamma$, where $|dz|$ denotes the

Euclidean element of length. (It is possible that $L\_\backslash rho(\backslash gamma)=\backslash infty$.)

What does this really mean?

If $\backslash gamma:I\backslash to\; D$ is parameterized in some interval $I$,

then $\backslash int\_\backslash gamma\; \backslash rho\backslash ,|dz|$ is the integral of the Borel-measurable function

$\backslash rho(\backslash gamma(t))$ with respect to the Borel measure on $I$

for which the measure of every subinterval $J\backslash subset\; I$ is the length of the

restriction of $\backslash gamma$ to $J$. In other words, it is the

Lebesgue-Stieltjes integral

$\backslash int\_I\; \backslash rho(\backslash gamma(t))\backslash ,d\{\backslash mathrm\{length\}\}\_\backslash gamma(t)$, where

$\{\backslash mathrm\{length\}\}\_\backslash gamma(t)$ is the length of the restriction of $\backslash gamma$

to $\backslash \{s\backslash in\; I:s\backslash le\; t\backslash \}$.

Also set

:$L\_\backslash rho(\backslash Gamma):=\backslash inf\_\{\backslash gamma\backslash in\backslash Gamma\}L\_\backslash rho(\backslash gamma).$

The area of $\backslash rho$ is defined as

:$A(\backslash rho):=\backslash int\_D\; \backslash rho^2\backslash ,dx\backslash ,dy,$

and the extremal length of $\backslash Gamma$ is

:$EL(\backslash Gamma):=\; \backslash sup\_\backslash rho\; \backslash frac\{L\_\backslash rho(\backslash Gamma)^2\}\{A(\backslash rho)\}\backslash ,,$

where the supremum is over all Borel-measureable $\backslash rho:D\backslash to[0,\backslash infty]$ with

$\backslash Gamma\_0$ denotes the set of rectifiable curves in $\backslash Gamma$, then

$EL(\backslash Gamma)$ is defined to be $EL(\backslash Gamma\_0)$.

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

The extremal distance in $D$ between two sets in $\backslash overline\; D$ 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.

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.

Fix some positive numbers $w,h>0$, and let $R$ be the rectangle $R=(0,w)\backslash times(0,h)$. Let $\backslash Gamma$ be the set of all finite length curves $\backslash gamma:(0,1)\backslash to\; R$ that cross the rectangle left to right, in the sense that $\backslash lim\_\{t\backslash to\; 0\}\backslash gamma(t)$

is on the left edge $\backslash \{0\backslash \}\backslash times[0,h]$ of the rectangle, and $\backslash lim\_\{t\backslash to\; 1\}\backslash gamma(t)$ is on the right edge $\backslash \{w\backslash \}\backslash times[0,h]$.

(The limits necessarily exist, because we are assuming that $\backslash gamma$ has finite length.) We will now prove that in this case

:$EL(\backslash Gamma)=w/h$

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

The opposite inequality is not quite so easy. Consider an arbitrary Borel-measurable $\backslash rho:R\backslash to[0,\backslash infty]$ such that

$\backslash ell:=L\_\backslash rho(\backslash Gamma)>0$.

For $y\backslash in(0,h)$, let $\backslash gamma\_y(t)=i\backslash ,y+w\backslash ,t$ (where we are identifying $\backslash R^2$ with the complex plane).

Then $\backslash gamma\_y\backslash in\backslash Gamma$, and hence $\backslash ell\backslash le\; L\_\backslash rho(\backslash gamma\_y)$.

The latter inequality may be written as

:$\backslash ell\backslash le\; \backslash int\_0^1\; \backslash rho(i\backslash ,y+w\backslash ,t)\backslash ,w\backslash ,dt\; .$

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

:$h\backslash ,\backslash ell\backslash le\; \backslash int\_0^h\backslash int\_0^1\backslash rho(i\backslash ,y+w\backslash ,t)\backslash ,w\backslash ,dt\backslash ,dy$.

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

:$h\backslash ,\backslash ell\; \backslash le\; \backslash int\_0^h\backslash int\_0^w\backslash rho(x+i\backslash ,y)\backslash ,dx\backslash ,dy\; \backslash le\; \backslash Bigl(\backslash int\_R\; \backslash rho^2\backslash ,dx\backslash ,dy\backslash int\_R\backslash ,dx\backslash ,dy\backslash Bigr)^\{1/2\}\; =\; \backslash bigl(w\backslash ,h\backslash ,A(\backslash rho)\backslash bigr)^\{1/2\}$. This gives $\backslash ell^2/A(\backslash rho)\backslash le\; w/h$.

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

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

It should be pointed out that the extremal length of the family of curves $\backslash Gamma\backslash ,\text{'}$ that connect the bottom edge of $R$ to the top edge of $R$ satisfies $EL(\backslash Gamma\backslash ,\text{'})=h/w$, by the same argument. Therefore, $EL(\backslash Gamma)\backslash ,EL(\backslash Gamma\backslash ,\text{'})=1$.

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 $EL(\backslash Gamma)$ is generally easier than obtaining an upper bound, since the lower bound involves choosing a reasonably good $\backslash rho$ and estimating $L\_\backslash rho(\backslash Gamma)^2/A(\backslash rho)$, while the upper bound involves proving a statement about all possible $\backslash rho$. For this reason, duality is often useful when it can be established: when we know that $EL(\backslash Gamma)\backslash ,EL(\backslash Gamma\backslash ,\text{'})=1$, a lower bound on $EL(\backslash Gamma\backslash ,\text{'})$ translates to an upper bound on $EL(\backslash Gamma)$.

Let $r\_1$ and $r\_2$ be two radii satisfying

To obtain a lower bound on $EL(\backslash Gamma)$, we take $\backslash rho(z)=1/|z|$. Then for $\backslash gamma\backslash in\backslash Gamma$ oriented from $C\_1$ to $C\_2$

:$\backslash int\_\backslash gamma\; |z|^\{-1\}\backslash ,ds\; \backslash ge\; \backslash int\_\backslash gamma\; |z|^\{-1\}\backslash ,d|z|\; =\; \backslash int\_\backslash gamma\; d\backslash log\; |z|=\backslash log(r\_2/r\_1).$

On the other hand,

:$A(\backslash rho)=\backslash int\_A\; |z|^\{-2\}\backslash ,dx\backslash ,dy=\; \backslash int\_\{0\}^\{2\backslash pi\}\backslash int\_\{r\_1\}^\{r\_2\}\; r^\{-2\}\backslash ,r\backslash ,dr\backslash ,d\backslash theta\; =\; 2\backslash ,\backslash pi\; \backslash ,\backslash log(r\_2/r\_1).$

We conclude that

:$EL(\backslash Gamma)\backslash ge\; \backslash frac\{\backslash log(r\_2/r\_1)\}\{2\backslash pi\}.$

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 $\backslash rho$ such that $\backslash ell:=L\_\backslash rho(\backslash Gamma)>0$. For $\backslash theta\backslash in[0,2\backslash ,\backslash pi)$ let $\backslash gamma\_\backslash theta:(r\_1,r\_2)\backslash to\; A$ denote the curve $\backslash gamma\_\backslash theta(r)=e^\{i\backslash theta\}r$. Then

:$\backslash ell\backslash le\backslash int\_\{\backslash gamma\_\backslash theta\}\backslash rho\backslash ,ds\; =\backslash int\_\{r\_1\}^\{r\_2\}\backslash rho(e^\{i\backslash theta\}r)\backslash ,dr.$

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

:$2\backslash ,\backslash pi\backslash ,\backslash ell\; \backslash le\; \backslash int\_A\; \backslash rho\backslash ,dr\backslash ,d\backslash theta\; \backslash le\; \backslash Bigl(\backslash int\_A\; \backslash rho^2\backslash ,r\backslash ,dr\backslash ,d\backslash theta\; \backslash Bigr)^\{1/2\}\backslash Bigl(\backslash int\_0^\{2\backslash pi\}\backslash int\_\{r\_1\}^\{r\_2\}\; \backslash frac\; 1\; r\backslash ,dr\backslash ,d\backslash theta\backslash Bigr)^\{1/2\}.$

Squaring gives

:$4\backslash ,\backslash pi^2\backslash ,\backslash ell^2\backslash le\; A(\backslash rho)\backslash cdot\backslash ,2\backslash ,\backslash pi\backslash ,\backslash log(r\_2/r\_1).$

This implies the upper bound $EL(\backslash Gamma)\backslash le\; (2\backslash ,\backslash pi)^\{-1\}\backslash ,\backslash log(r\_2/r\_1)$.

When combined with the lower bound, this yields the exact value of the extremal length:

:$EL(\backslash Gamma)=\backslash frac\{\backslash log(r\_2/r\_1)\}\{2\backslash pi\}.$

Let $r\_1,r\_2,C\_1,C\_2,\backslash Gamma$ and $A$ be as above, but now let $\backslash Gamma^*$ 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

:$EL(\backslash Gamma^*)=\backslash frac\{2\backslash pi\}\{\backslash log(r\_2/r\_1)\}=EL(\backslash Gamma)^\{-1\}.$

This illustrates another instance of extremal length duality.

In the above examples, the extremal $\backslash rho$ which maximized the ratio $L\_\backslash rho(\backslash Gamma)^2/A(\backslash rho)$ 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 $\backslash rho$, 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 $\backslash R^3$ with its Riemannian spherical metric. In other words, this is the quotient of the sphere by the map $x\backslash mapsto\; -x$. Let $\backslash Gamma$ denote the set of closed curves in this projective plane that are not null-homotopic. (Each curve in $\backslash Gamma$ 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.Ahlfors (1973) (The definition of extremal length readily extends to Riemannian surfaces.) Thus, the extremal length is $\backslash pi^2/(2\backslash ,\backslash pi)=\backslash pi/2$.

If $\backslash Gamma$ is any collection of paths all of which have positive diameter and containing a point $z\_0$, then $EL(\backslash Gamma)=\backslash infty$. This follows, for example, by taking

:

0 & |z-z_0|\ge 1/2,\end{cases} which satisfies and $L\_\backslash rho(\backslash gamma)=\backslash infty$ for every rectifiable $\backslash gamma\backslash in\backslash Gamma$.

The extremal length satisfies a few simple monotonicity properties. First, it is clear that if $\backslash Gamma\_1\backslash subset\backslash Gamma\_2$, then $EL(\backslash Gamma\_1)\backslash ge\; EL(\backslash Gamma\_2)$.

Moreover, the same conclusion holds if every curve $\backslash gamma\_1\backslash in\backslash Gamma\_1$ contains a curve $\backslash gamma\_2\backslash in\; \backslash Gamma\_2$ as a subcurve (that is, $\backslash gamma\_2$ is the restriction of $\backslash gamma\_1$ to a subinterval of its domain). Another sometimes useful inequality is

:$EL(\backslash Gamma\_1\backslash cup\backslash Gamma\_2)\backslash ge\; \backslash bigl(EL(\backslash Gamma\_1)^\{-1\}+EL(\backslash Gamma\_2)^\{-1\}\backslash bigr)^\{-1\}.$

This is clear if $EL(\backslash Gamma\_1)=0$ or if $EL(\backslash Gamma\_2)=0$, 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 $\backslash Gamma\_1\backslash cup\backslash Gamma\_2$ are all rectifiable. Let $\backslash rho\_1,\backslash rho\_2$ satisfy $L\_\{\backslash rho\_j\}(\backslash Gamma\_j)\backslash ge\; 1$ for $j=1,2$. Set $\backslash rho=\backslash max\backslash \{\backslash rho\_1,\backslash rho\_2\backslash \}$. Then $L\_\backslash rho(\backslash Gamma\_1\backslash cup\backslash Gamma\_2)\backslash ge\; 1$ and $A(\backslash rho)=\backslash int\backslash rho^2\backslash ,dx\backslash ,dy\backslash le\backslash int(\backslash rho\_1^2+\backslash rho\_2^2)\backslash ,dx\backslash ,dy=A(\backslash rho\_1)+A(\backslash rho\_2)$, which proves the inequality.

Let $f:D\backslash to\; D^*$ be a conformal homeomorphism

(a bijective holomorphic map) between planar domains. Suppose that

$\backslash Gamma$ is a collection of curves in $D$,

and let $\backslash Gamma^*:=\backslash \{f\backslash circ\; \backslash gamma:\backslash gamma\backslash in\backslash Gamma\backslash \}$ denote the

image curves under $f$. Then $EL(\backslash Gamma)=EL(\backslash Gamma^*)$.

This conformal invariance statement is the primary reason why the concept of

extremal length is useful.

Here is a proof of conformal invariance. Let $\backslash Gamma\_0$ denote the set of curves

$\backslash gamma\backslash in\backslash Gamma$ such that $f\backslash circ\; \backslash gamma$ is rectifiable, and let

$\backslash Gamma\_0^*=\backslash \{f\backslash circ\backslash gamma:\backslash gamma\backslash in\backslash Gamma\_0\backslash \}$, which is the set of rectifiable

curves in $\backslash Gamma^*$. Suppose that $\backslash rho^*:D^*\backslash to[0,\backslash infty]$ is Borel-measurable. Define

:$\backslash rho(z)=|f\backslash ,\text{'}(z)|\backslash ,\backslash rho^*\backslash bigl(f(z)\backslash bigr).$

A change of variables $w=f(z)$ gives

:$A(\backslash rho)=\backslash int\_D\; \backslash rho(z)^2\backslash ,dz\backslash ,d\backslash bar\; z=\backslash int\_D\; \backslash rho^*(f(z))^2\backslash ,|f\backslash ,\text{'}(z)|^2\backslash ,dz\backslash ,d\backslash bar\; z\; =\; \backslash int\_\{D^*\}\; \backslash rho^*(w)^2\backslash ,dw\backslash ,d\backslash bar\; w=A(\backslash rho^*).$

Now suppose that $\backslash gamma\backslash in\; \backslash Gamma\_0$ is rectifiable, and set $\backslash gamma^*:=f\backslash circ\backslash gamma$. Formally, we may use a change of variables again:

:$L\_\backslash rho(\backslash gamma)=\backslash int\_\backslash gamma\; \backslash rho^*\backslash bigl(f(z)\backslash bigr)\backslash ,|f\backslash ,\text{'}(z)|\backslash ,|dz|\; =\; \backslash int\_\{\backslash gamma^*\}\; \backslash rho(w)\backslash ,|dw|=L\_\{\backslash rho^*\}(\backslash gamma^*).$

To justify this formal calculation, suppose that $\backslash gamma$ is defined in some interval $I$, let

$\backslash ell(t)$ denote the length of the restriction of $\backslash gamma$ to $I\backslash cap(-\backslash infty,t]$,

and let $\backslash ell^*(t)$ be similarly defined with $\backslash gamma^*$ in place of $\backslash gamma$. Then it is easy to see that $d\backslash ell^*(t)=|f\backslash ,\text{'}(\backslash gamma(t))|\backslash ,d\backslash ell(t)$, and this implies $L\_\backslash rho(\backslash gamma)=L\_\{\backslash rho^*\}(\backslash gamma^*)$, as required. The above equalities give,

:$EL(\backslash Gamma\_0)\backslash ge\; EL(\backslash Gamma\_0^*)=EL(\backslash Gamma^*).$

If we knew that each curve in $\backslash Gamma$ and $\backslash Gamma^*$ was rectifiable, this would

prove $EL(\backslash Gamma)=EL(\backslash Gamma^*)$ since we may also apply the above with $f$ replaced by its inverse

and $\backslash Gamma$ interchanged with $\backslash Gamma^*$. It remains to handle the non-rectifiable curves.

Now let $\backslash hat\backslash Gamma$ denote the set of rectifiable curves $\backslash gamma\backslash in\backslash Gamma$ such that $f\backslash circ\backslash gamma$ is

non-rectifiable. We claim that $EL(\backslash hat\backslash Gamma)=\backslash infty$.

Indeed, take $\backslash rho(z)=|f\backslash ,\text{'}(z)|\backslash ,h(|f(z)|)$, where $h(r)=\backslash bigl(r\backslash ,\backslash log\; (r+2)\backslash bigr)^\{-1\}$.

Then a change of variable as above gives

:

For $\backslash gamma\backslash in\backslash hat\backslash Gamma$ and $r\backslash in(0,\backslash infty)$ such that $f\backslash circ\; \backslash gamma$

is contained in

:$L\_\backslash rho(\backslash gamma)\backslash ge\backslash inf\backslash \{h(s):s\backslash in[0,r]\backslash \}\backslash ,\backslash mathrm\{length\}(f\backslash circ\backslash gamma)=\backslash infty$.{{Dubious|date=November 2008}}

On the other hand, suppose that $\backslash gamma\backslash in\backslash hat\backslash Gamma$ is such that $f\backslash circ\backslash gamma$ is unbounded.

Set $H(t):=\backslash int\_0^t\; h(s)\backslash ,ds$. Then

$L\_\backslash rho(\backslash gamma)$ is at least the length of the curve $t\backslash mapsto\; H(|f\backslash circ\; \backslash gamma(t)|)$

(from an interval in $\backslash R$ to $\backslash R$). Since $\backslash lim\_\{t\backslash to\backslash infty\}H(t)=\backslash infty$,

it follows that $L\_\backslash rho(\backslash gamma)=\backslash infty$.

Thus, indeed, $EL(\backslash hat\backslash Gamma)=\backslash infty$.

Using the results of the previous section, we have

:$EL(\backslash Gamma)=EL(\backslash Gamma\_0\backslash cup\backslash hat\backslash Gamma)\backslash ge\; EL(\backslash Gamma\_0)$.

We have already seen that $EL(\backslash Gamma\_0)\backslash ge\; EL(\backslash Gamma^*)$. Thus, $EL(\backslash Gamma)\backslash ge\; EL(\backslash Gamma^*)$.

The reverse inequality holds by symmetry, and conformal invariance is therefore established.

==Some applications of extremal length==

By the calculation of the extremal distance in an annulus and the conformal

invariance it follows that the annulus

is not conformally homeomorphic to the annulus

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

{{Expand section|date=June 2008}}

Suppose that $G=(V,E)$ is some graph and $\backslash Gamma$ 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,Duffin 1962 consider a function $\backslash rho:E\backslash to[0,\backslash infty)$. The $\backslash rho$-length of a path is defined as the sum of $\backslash rho(e)$ over all edges in the path, counted with multiplicity. The "area" $A(\backslash rho)$ is defined as $\backslash sum\_\{e\backslash in\; E\}\backslash rho(e)^2$. The extremal length of $\backslash Gamma$ 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 vertices 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 $\backslash rho:V\backslash to[0,\backslash infty)$, the area is $A(\backslash rho):=\backslash sum\_\{v\backslash in\; V\}\backslash rho(v)^2$, and the length of a path is the sum of $\backslash rho(v)$ over the vertices visited by the path, with multiplicity.

{{reflist|2}}

• {{Citation | author1-link=Lars Ahlfors | last1=Ahlfors | first1=Lars V. | title=Conformal invariants: topics in geometric function theory | publisher=McGraw-Hill Book Co. | location=New York | mr=0357743 | year=1973}}
• {{Citation | last1=Duffin | first1=R. J. | title= The extremal length of a network | year=1962 | journal=Journal of Mathematical Analysis and Applications | volume=5 | pages=200–215 | doi=10.1016/S0022-247X(62)80004-3 | issue=2| doi-access=free }}
• {{Citation | last1=Lehto | first1=O. | last2=Virtanen | first2=K. I. | title=Quasiconformal mappings in the plane | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | year=1973}}

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Category:Conformal mappings

Category:Potential theory