In the mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 theory of conformal and quasiconformal mapping

Quasiconformal mapping

In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity....

s, the extremal length of a collection of curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s is a conformal invariant of . More specifically, suppose that
is an open set in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 and is a collection
of paths in and is a conformal mapping. Then the extremal length of is equal to the extremal length of the image of under . 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

To define extremal length, we need to first introduce several related quantities.
Let be an open set in the complex plane. Suppose that is a
collection of rectifiable curves in . If
is Borel-measurable

Borel algebra

In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...

, then for any rectifiable curve we let


denote the -length of , where denotes the
Euclidean

Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

 element of length. (It is possible that .)
What does this really mean?
If is parameterized in some interval ,
then is the integral of the Borel-measurable function
with respect to the Borel measure on
for which the measure of every subinterval is the length of the
restriction of to . In other words, it is the
Lebesgue-Stieltjes integral

Lebesgue-Stieltjes integration

In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework...

, where
is the length of the restriction of
to .
Also set


The area of is defined as
and the extremal length of is


where the supremum is over all Borel-measureable with . If contains some non-rectifiable curves and
denotes the set of rectifiable curves in , then
is defined to be .

The term modulus of refers to .

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

Examples

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

Fix some positive numbers , and let be the rectangle
. Let be the set of all finite
length curves that cross the rectangle left to right,
in the sense that
is on the left edge of the rectangle, and
is on the right edge .
(The limits necessarily exist, because we are assuming that
has finite length.) We will now prove that in this case

First, we may take on . This
gives and . The definition
of as a supremum then gives .

The opposite inequality is not quite so easy. Consider an arbitrary
Borel-measurable such that
.
For , let
(where we are identifying with the complex plane).
Then , and hence .
The latter inequality may be written as
Integrating this inequality over implies.
Now a change of variable and an application of the Cauchy-Schwarz inequality give. This gives .
Therefore, , as required.

As the proof shows, the extremal length of is the same as the extremal
length of the much smaller collection of curves .

It should be pointed out that the extremal length of the family of curves
that connect the bottom edge of to the top edge of satisfies
, by the same argument. Therefore, .
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
is generally easier than obtaining an upper bound, since the lower bound involves
choosing a reasonably good and estimating ,
while the upper bound involves proving a statement about all possible . For this reason,
duality is often useful when it can be established: when we know that ,
a lower bound on translates to an upper bound on .

Extremal distance in annulus

Let and be two radii satisfying
. Let be the
annulus and let
and be the two boundary components
of :
and . Consider the extremal distance
in between and ;
which is the extremal length of the collection of
curves connecting
and .

To obtain an lower bound on ,
we take . Then for
oriented from to
On the other hand,
We conclude that

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 such that . For let denote the curve . Then
We integrate over and apply the Cauchy-Schwarz inequality, to obtain:
Squaring gives
This implies the upper bound .
When combined with the lower bound, this yields the exact value of the extremal length:

Extremal length around an annulus

Let and be as above, but now let be the collection of all curves that wind once around the annulus, separating from . Using the above methods, it is not hard to show that
This illustrates another instance of extremal length duality.

Extremal length of topologically essential paths in projective plane

In the above examples, the extremal which maximized the
ratio and gave the extremal length corresponded to a flat metric. In other words, when the Euclidean

Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

 Riemannian metric of the corresponding planar domain is scaled by , 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

Cylinder (geometry)

A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called 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 point

Antipodal point

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....

s on the unit sphere in with its Riemannian spherical metric. In other words, this is the quotient of the sphere by the map . Let denote the set of closed curves in this projective plane that are not null-homotopic. (Each curve in 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. (The definition of extremal length readily extends to Riemannian surfaces.) Thus, the extremal length is .

Extremal length of paths containing a point

If is any collection of paths all of which have positive diameter and containing a point , then . This follows, for example, by taking
which satisfies and for every rectifiable .

Elementary properties of extremal length

The extremal length satisfies a few simple monotonicity properties. First, it is clear that if , then .
Moreover, the same conclusion holds if every curve contains a curve as a subcurve (that is, is the restriction of to a subinterval of its domain). Another sometimes useful inequality is
This is clear if or if , in which case the right hand side is interpreted as . So suppose that this is not the case and with no loss of generality assume that the curves in are all rectifiable. Let satisfy for . Set . Then and , which proves the inequality.

Conformal invariance of extremal length

Let be a conformal

Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

 homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

(a bijective holomorphic map) between planar domains. Suppose that
is a collection of curves in ,
and let denote the
image curves under . Then .
This conformal invariance statement is the primary reason why the concept of
extremal length is useful.

Here is a proof of conformal invariance. Let denote the set of curves
such that is rectifiable, and let
, which is the set of rectifiable
curves in . Suppose that is Borel-measurable. Define
A change of variables gives
Now suppose that is rectifiable, and set . Formally, we may use a change of variables again:
To justify this formal calculation, suppose that is defined in some interval , let
denote the length of the restriction of to ,
and let be similarly defined with in place of . Then it is easy to see that , and this implies , as required. The above equalities give,
If we knew that each curve in and was rectifiable, this would
prove since we may also apply the above with replaced by its inverse
and interchanged with . It remains to handle the non-rectifiable curves.

Now let denote the set of rectifiable curves such that is
non-rectifiable. We claim that .
Indeed, take , where .
Then a change of variable as above gives
For and such that
is contained in , we have.
On the other hand, suppose that is such that is unbounded.
Set . Then
is at least the length of the curve
(from an interval in to ). Since ,
it follows that .
Thus, indeed, .

Using the results of the previous section, we have.
We have already seen that . Thus, .
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 (where )
is not conformally homeomorphic to the annulus if .

Extremal length in higher dimensions

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

Suppose that is some graph

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

 and is a collection of paths in . There are two variants of extremal length in this setting. To define the edge extremal length, originally introduced by R. J. Duffin, consider a function . The -length of a path is defined as the sum of over all edges in the path, counted with multiplicity. The "area" is defined as . The extremal length of is then defined as before. If 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

Potential theory

In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

.

Another notion of discrete extremal length that is appropriate in other contexts is vertex extremal length, where , the area is , and the length of a path is the sum of over the vertices visited by the path, with multiplicity.