Extremal length

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In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves Γ is a measure of the size of Γ that is invariant under conformal mappings. More specifically, suppose that D is an open set in the complex plane and Γ is a collection of paths in D and f:DD is a conformal mapping. Then the extremal length of Γ is equal to the extremal length of the image of Γ under f. One also works with the conformal modulus of Γ, the reciprocal of the extremal length. The fact that extremal length and conformal modulus are conformal invariants of Γ 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 D be an open set in the complex plane. Suppose that Γ is a collection of rectifiable curves in D. If ρ:D[0,] is Borel-measurable, then for any rectifiable curve γ we let

Lρ(γ):=γρ|dz|

denote the ρ–length of γ, where |dz| denotes the Euclidean element of length. (It is possible that Lρ(γ)=.) What does this really mean? If γ:ID is parameterized in some interval I, then γρ|dz| is the integral of the Borel-measurable function ρ(γ(t)) with respect to the Borel measure on I for which the measure of every subinterval JI is the length of the restriction of γ to J. In other words, it is the Lebesgue-Stieltjes integral Iρ(γ(t))dlengthγ(t), where lengthγ(t) is the length of the restriction of γ to {sI:st}. Also set

Lρ(Γ):=infγΓLρ(γ).

The area of ρ is defined as

A(ρ):=Dρ2dxdy,

and the extremal length of Γ is

EL(Γ):=supρLρ(Γ)2A(ρ),

where the supremum is over all Borel-measureable ρ:D[0,] with 0<A(ρ)<. If Γ contains some non-rectifiable curves and Γ0 denotes the set of rectifiable curves in Γ, then EL(Γ) is defined to be EL(Γ0).

The term (conformal) modulus of Γ refers to 1/EL(Γ).

The extremal distance in D between two sets in 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

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)×(0,h). Let Γ be the set of all finite length curves γ:(0,1)R that cross the rectangle left to right, in the sense that limt0γ(t) is on the left edge {0}×[0,h] of the rectangle, and limt1γ(t) is on the right edge {w}×[0,h]. (The limits necessarily exist, because we are assuming that γ has finite length.) We will now prove that in this case

EL(Γ)=w/h

First, we may take ρ=1 on R. This ρ gives A(ρ)=wh and Lρ(Γ)=w. The definition of EL(Γ) as a supremum then gives EL(Γ)w/h.

The opposite inequality is not quite so easy. Consider an arbitrary Borel-measurable ρ:R[0,] such that :=Lρ(Γ)>0. For y(0,h), let γy(t)=iy+wt (where we are identifying 2 with the complex plane). Then γyΓ, and hence Lρ(γy). The latter inequality may be written as

01ρ(iy+wt)wdt.

Integrating this inequality over y(0,h) implies

h0h01ρ(iy+wt)wdtdy.

Now a change of variable x=wt and an application of the Cauchy–Schwarz inequality give

h0h0wρ(x+iy)dxdy(Rρ2dxdyRdxdy)1/2=(whA(ρ))1/2. This gives 2/A(ρ)w/h.

Therefore, EL(Γ)w/h, as required.

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

It should be pointed out that the extremal length of the family of curves Γ that connect the bottom edge of R to the top edge of R satisfies EL(Γ)=h/w, by the same argument. Therefore, EL(Γ)EL(Γ)=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(Γ) is generally easier than obtaining an upper bound, since the lower bound involves choosing a reasonably good ρ and estimating Lρ(Γ)2/A(ρ), while the upper bound involves proving a statement about all possible ρ. For this reason, duality is often useful when it can be established: when we know that EL(Γ)EL(Γ)=1, a lower bound on EL(Γ) translates to an upper bound on EL(Γ).

Extremal distance in annulus

Let r1 and r2 be two radii satisfying 0<r1<r2<. Let A be the annulus A:={z:r1<|z|<r2} and let C1 and C2 be the two boundary components of A: C1:={z:|z|=r1} and C2:={z:|z|=r2}. Consider the extremal distance in A between C1 and C2; which is the extremal length of the collection Γ of curves γA connecting C1 and C2.

To obtain a lower bound on EL(Γ), we take ρ(z)=1/|z|. Then for γΓ oriented from C1 to C2

γ|z|1dsγ|z|1d|z|=γdlog|z|=log(r2/r1).

On the other hand,

A(ρ)=A|z|2dxdy=02πr1r2r2rdrdθ=2πlog(r2/r1).

We conclude that

EL(Γ)log(r2/r1)2π.

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 ρ such that :=Lρ(Γ)>0. For θ[0,2π) let γθ:(r1,r2)A denote the curve γθ(r)=eiθr. Then

γθρds=r1r2ρ(eiθr)dr.

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

2πAρdrdθ(Aρ2rdrdθ)1/2(02πr1r21rdrdθ)1/2.

Squaring gives

4π22A(ρ)2πlog(r2/r1).

This implies the upper bound EL(Γ)(2π)1log(r2/r1). When combined with the lower bound, this yields the exact value of the extremal length:

EL(Γ)=log(r2/r1)2π.

Extremal length around an annulus

Let r1,r2,C1,C2,Γ and A be as above, but now let Γ* be the collection of all curves that wind once around the annulus, separating C1 from C2. Using the above methods, it is not hard to show that

EL(Γ*)=2πlog(r2/r1)=EL(Γ)1.

This illustrates another instance of extremal length duality.

Extremal length of topologically essential paths in projective plane

In the above examples, the extremal ρ which maximized the ratio Lρ(Γ)2/A(ρ) 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 ρ, 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 3 with its Riemannian spherical metric. In other words, this is the quotient of the sphere by the map xx. Let Γ denote the set of closed curves in this projective plane that are not null-homotopic. (Each curve in Γ 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 π2/(2π)=π/2.

Extremal length of paths containing a point

If Γ is any collection of paths all of which have positive diameter and containing a point z0, then EL(Γ)=. This follows, for example, by taking

ρ(z):={(|zz0|log|zz0|)1|zz0|<1/2,0|zz0|1/2, which satisfies A(ρ)< and Lρ(γ)= for every rectifiable γΓ.

Elementary properties of extremal length

The extremal length satisfies a few simple monotonicity properties. First, it is clear that if Γ1Γ2, then EL(Γ1)EL(Γ2). Moreover, the same conclusion holds if every curve γ1Γ1 contains a curve γ2Γ2 as a subcurve (that is, γ2 is the restriction of γ1 to a subinterval of its domain). Another sometimes useful inequality is

EL(Γ1Γ2)(EL(Γ1)1+EL(Γ2)1)1.

This is clear if EL(Γ1)=0 or if EL(Γ2)=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 Γ1Γ2 are all rectifiable. Let ρ1,ρ2 satisfy Lρj(Γj)1 for j=1,2. Set ρ=max{ρ1,ρ2}. Then Lρ(Γ1Γ2)1 and A(ρ)=ρ2dxdy(ρ12+ρ22)dxdy=A(ρ1)+A(ρ2), which proves the inequality.

Conformal invariance of extremal length

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

Here is a proof of conformal invariance. Let Γ0 denote the set of curves γΓ such that fγ is rectifiable, and let Γ0*={fγ:γΓ0}, which is the set of rectifiable curves in Γ*. Suppose that ρ*:D*[0,] is Borel-measurable. Define

ρ(z)=|f(z)|ρ*(f(z)).

A change of variables w=f(z) gives

A(ρ)=Dρ(z)2dzdz¯=Dρ*(f(z))2|f(z)|2dzdz¯=D*ρ*(w)2dwdw¯=A(ρ*).

Now suppose that γΓ0 is rectifiable, and set γ*:=fγ. Formally, we may use a change of variables again:

Lρ(γ)=γρ*(f(z))|f(z)||dz|=γ*ρ(w)|dw|=Lρ*(γ*).

To justify this formal calculation, suppose that γ is defined in some interval I, let (t) denote the length of the restriction of γ to I(,t], and let *(t) be similarly defined with γ* in place of γ. Then it is easy to see that d*(t)=|f(γ(t))|d(t), and this implies Lρ(γ)=Lρ*(γ*), as required. The above equalities give,

EL(Γ0)EL(Γ0*)=EL(Γ*).

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

Now let Γ^ denote the set of rectifiable curves γΓ such that fγ is non-rectifiable. We claim that EL(Γ^)=. Indeed, take ρ(z)=|f(z)|h(|f(z)|), where h(r)=(rlog(r+2))1. Then a change of variable as above gives

A(ρ)=D*h(|w|)2dwdw¯02π0(rlog(r+2))2rdrdθ<.

For γΓ^ and r(0,) such that fγ is contained in {z:|z|<r}, we have

Lρ(γ)inf{h(s):s[0,r]}length(fγ)=.[dubious ]

On the other hand, suppose that γΓ^ is such that fγ is unbounded. Set H(t):=0th(s)ds. Then Lρ(γ) is at least the length of the curve tH(|fγ(t)|) (from an interval in to ). Since limtH(t)=, it follows that Lρ(γ)=. Thus, indeed, EL(Γ^)=.

Using the results of the previous section, we have

EL(Γ)=EL(Γ0Γ^)EL(Γ0).

We have already seen that EL(Γ0)EL(Γ*). Thus, EL(Γ)EL(Γ*). 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 {z:r<|z|<R} (where 0r<R) is not conformally homeomorphic to the annulus {w:r*<|w|<R*} if RrR*r*.

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 and Γ 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 ρ:E[0,). The ρ-length of a path is defined as the sum of ρ(e) over all edges in the path, counted with multiplicity. The "area" A(ρ) is defined as eEρ(e)2. The extremal length of Γ 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 ρ:V[0,), the area is A(ρ):=vVρ(v)2, and the length of a path is the sum of ρ(v) over the vertices visited by the path, with multiplicity.

Notes

  1. Ahlfors (1973)
  2. Duffin 1962

References

  • Ahlfors, Lars V. (1973), Conformal invariants: topics in geometric function theory, New York: McGraw-Hill Book Co. 
  • 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