In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves
is a measure of the size of
that is invariant under conformal mappings. More specifically, suppose that
is an open set in the complex plane 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
. One also works with the conformal modulus of
, the reciprocal of the extremal length. The fact that extremal length and conformal modulus are conformal invariants of
makes them useful tools in the study of conformal and quasi-conformal mappings. One also works with extremal length in dimensions greater than two and certain other metric spaces, 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, then for any rectifiable curve
we let
![{\displaystyle L_{\rho }(\gamma ):=\int _{\gamma }\rho \,|dz|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f853b266c802ad7ce536ebfa6eba5d4b68b3faca)
denote the
–length of
, where
denotes the Euclidean 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
, where
is the length of the restriction of
to
. Also set
![{\displaystyle L_{\rho }(\Gamma ):=\inf _{\gamma \in \Gamma }L_{\rho }(\gamma ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fee1c30c37a7823c7f71736f737d0286c88b310)
The area of
is defined as
![{\displaystyle A(\rho ):=\int _{D}\rho ^{2}\,dx\,dy,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1061d5d49f59c9e7fa5865b157a492479beb478)
and the extremal length of
is
![{\displaystyle EL(\Gamma ):=\sup _{\rho }{\frac {L_{\rho }(\Gamma )^{2}}{A(\rho )}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48dc39cc5845204960518dd4da803c2185ba3e2f)
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 (conformal) 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
![{\displaystyle EL(\Gamma )=w/h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d63c6c958db6bc9080b6d66a6421276e043d410)
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
![{\displaystyle \ell \leq \int _{0}^{1}\rho (i\,y+w\,t)\,w\,dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f97ef6436932e9a2d94330ccbc2ca6418a7be856)
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 a lower bound on
, we take
. Then for
oriented from
to
![{\displaystyle \int _{\gamma }|z|^{-1}\,ds\geq \int _{\gamma }|z|^{-1}\,d|z|=\int _{\gamma }d\log |z|=\log(r_{2}/r_{1}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6429fbb455cabd261d7a6cdba4399c7c5adbcc03)
On the other hand,
![{\displaystyle 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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a11fc8530070570bc6a690dda033245d02dd598)
We conclude that
![{\displaystyle EL(\Gamma )\geq {\frac {\log(r_{2}/r_{1})}{2\pi }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1b57f79e38dc13828c05a0903bd0d6b3b20603)
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
![{\displaystyle \ell \leq \int _{\gamma _{\theta }}\rho \,ds=\int _{r_{1}}^{r_{2}}\rho (e^{i\theta }r)\,dr.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5cc8ec29f59ba954ce3a508e288925ee84b443f)
We integrate over
and apply the Cauchy-Schwarz inequality, to obtain:
![{\displaystyle 2\,\pi \,\ell \leq \int _{A}\rho \,dr\,d\theta \leq {\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d86bad0fec5e2e8b076394a5b7b81d5cb933f8c5)
Squaring gives
![{\displaystyle 4\,\pi ^{2}\,\ell ^{2}\leq A(\rho )\cdot \,2\,\pi \,\log(r_{2}/r_{1}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2db4f5e336803db898b66083d24b2f099a687ab0)
This implies the upper bound
. When combined with the lower bound, this yields the exact value of the extremal length:
![{\displaystyle EL(\Gamma )={\frac {\log(r_{2}/r_{1})}{2\pi }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de2a5a5589e30ccedf16ea9bd5e88ea682b324d9)
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
![{\displaystyle EL(\Gamma ^{*})={\frac {2\pi }{\log(r_{2}/r_{1})}}=EL(\Gamma )^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eac1c6e4d41bfdb6148724e5c26585c3870c6ddb)
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 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. 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
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.[1] (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
![{\displaystyle EL(\Gamma _{1}\cup \Gamma _{2})\geq {\bigl (}EL(\Gamma _{1})^{-1}+EL(\Gamma _{2})^{-1}{\bigr )}^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd82584c3769d2a72dcfd405ba140c0c647eec93)
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 homeomorphism (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
![{\displaystyle \rho (z)=|f\,'(z)|\,\rho ^{*}{\bigl (}f(z){\bigr )}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd0448a90725acf1e75efb445cb0d3be6c2f5d3)
A change of variables
gives
![{\displaystyle 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 ^{*}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d5355d9d592375282d09aba82116d88590f92c)
Now suppose that
is rectifiable, and set
. Formally, we may use a change of variables again:
![{\displaystyle L_{\rho }(\gamma )=\int _{\gamma }\rho ^{*}{\bigl (}f(z){\bigr )}\,|f\,'(z)|\,|dz|=\int _{\gamma ^{*}}\rho (w)\,|dw|=L_{\rho ^{*}}(\gamma ^{*}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cbae4fa1df017fdf6807bda72891faffafc843b)
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,
![{\displaystyle EL(\Gamma _{0})\geq EL(\Gamma _{0}^{*})=EL(\Gamma ^{*}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f85d1bcb1b2d41580a97158b17c4f5d6ff052efc)
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
![{\displaystyle A(\rho )=\int _{D^{*}}h(|w|)^{2}\,dw\,d{\bar {w}}\leq \int _{0}^{2\pi }\int _{0}^{\infty }(r\,\log(r+2))^{-2}\,r\,dr\,d\theta <\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e59e31235710fe9757a92827a004c772dacf4294)
For
and
such that
is contained in
, we have
.[dubious – discuss]
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 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,[2] 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 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
, the area is
, and the length of a path is the sum of
over the vertices visited by the path, with multiplicity.
Notes
- ^ Ahlfors (1973)
- ^ Duffin 1962
References
- 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