In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
theory of
conformal
Conformal may refer to:
* Conformal (software), in ASIC Software
* Conformal coating in electronics
* Conformal cooling channel, in injection or blow moulding
* Conformal field theory in physics, such as:
** Boundary conformal field theory ...
and
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.
Intuitively, let ''f'' : '' ...
s, the extremal length of a collection of
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s
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
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
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 invariant
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s 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 space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
s, 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 curve
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Rec ...
s in
. If
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
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_\gamma(t)
is on the left edge
\\times ,h/math> of the rectangle, and \lim_\gamma(t) is on the right edge \\times ,h/math>.
(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 ,\infty/math> 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
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality f ...
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)^ = \bigl(w\,h\,A(\rho)\bigr)^. 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
\.
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
Let
r_1 and
r_2 be two radii satisfying
0. Let A be the annulus A:=\ and let C_1 and C_2 be the two boundary components of A: C_1:=\ and C_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, ^\,ds \ge \int_\gamma , z, ^\,d, z, = \int_\gamma d\log , z, =\log(r_2/r_1).
On the other hand,
:A(\rho)=\int_A , z, ^\,dx\,dy= \int_^\int_^ r^\,r\,dr\,d\theta = 2\,\pi \,\log(r_2/r_1).
We conclude that
:EL(\Gamma)\ge \frac.
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^r. Then
:\ell\le\int_\rho\,ds =\int_^\rho(e^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)^\Bigl(\int_0^\int_^ \frac 1 r\,dr\,d\theta\Bigr)^.
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)^\,\log(r_2/r_1).
When combined with the lower bound, this yields the exact value of the extremal length:
:EL(\Gamma)=\frac.
Extremal length around an annulus
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=EL(\Gamma)^.
This illustrates another instance of extremal length duality.
Extremal length of topologically essential paths in projective plane
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
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infi ...
. 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
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
s 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. (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
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(-, z-z_0, \,\log , z-z_0, )^ & , z-z_0, <1/2,\\
0 & , z-z_0, \ge 1/2,\end which satisfies
A(\rho)<\infty and
L_\rho(\gamma)=\infty for every rectifiable
\gamma\in\Gamma.
Elementary properties of extremal length
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)^+EL(\Gamma_2)^\bigr)^.
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_(\Gamma_j)\ge 1 for
j=1,2. Set
\rho=\max\. 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
Let
f:D\to D^* be a
conformal
Conformal may refer to:
* Conformal (software), in ASIC Software
* Conformal coating in electronics
* Conformal cooling channel, in injection or blow moulding
* Conformal field theory in physics, such as:
** Boundary conformal field theory ...
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
(a
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
holomorphic map) between planar domains. Suppose that
\Gamma is a collection of curves in
D,
and let
\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^*=\, which is the set of rectifiable
curves in
\Gamma^*. Suppose that
\rho^*:D^*\to ,\infty/math> 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_ \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_ \rho(w)\,, dw, =L_(\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_(\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)^.
Then a change of variable as above gives
:A(\rho)= \int_ h(, w, )^2\,dw\,d\bar w \le \int_0^\int_0^\infty (r\,\log (r+2))^ \,r\,dr\,d\theta<\infty.
For \gamma\in\hat\Gamma and r\in(0,\infty) such that f\circ \gamma
is contained in \, we have
:L_\rho(\gamma)\ge\inf\\,\mathrm(f\circ\gamma)=\infty.
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_H(t)=\infty,
it follows that L_\rho(\gamma)=\infty.
Thus, indeed, EL(\hat\Gamma)=\infty.
Using the results of the #Elementary properties of extremal length, 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
By the
calculation
A calculation is a deliberate mathematical process that transforms one or more inputs into one or more outputs or ''results''. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to t ...
of the extremal distance in an annulus and the conformal
invariance it follows that the annulus
\ (where
0\le r)
is not conformally homeomorphic to the annulus \ if \frac Rr\ne \frac.
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
G=(V,E) is some
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
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
R. or r. may refer to:
* '' Reign'', the period of time during which an Emperor, king, queen, etc., is ruler.
* '' Rex'', abbreviated as R., the Latin word meaning King
* ''Regina'', abbreviated as R., the Latin word meaning Queen
* or , abbrevi ...
,
[Duffin 1962] 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_\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 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
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
.
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_\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
References
*
*
*
{{DEFAULTSORT:Extremal Length
Conformal mappings
Potential theory