Extremal length

In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves $$Gamma is a conformal invariant of $$Gamma. More specifically, suppose that$$D is an open set in the complex plane and $$Gamma is a collectionof paths in $$D and $$f:D o D' is a conformal mapping. Then the extremal length of $$Gamma is equal to the extremal length of the image of $$Gamma under $$f. 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 $$D be an open set in the complex plane. Suppose that $$Gamma is acollection of rectifiable curves in $$D. If $$ho:D o [0,infty] is Borel-measurable, then for any rectifiable curve $$gamma we let

:$$L_ ho(gamma):=int_gamma ho,|dz|

denote the $$ho-length of $$gamma, where $$|dz| denotes the
Euclidean element of length. (It is possible that $$L_ ho(gamma)=infty.)What does this really mean? If $$gamma:I o D is parameterized in some interval $$I,then $$int_gamma ho,|dz| is the integral of the Borel-measurable function$$ho(gamma(t)) with respect to the Borel measure on $$Ifor which the measure of every subinterval $$Jsubset I is the length of therestriction of $$gamma to $$J. In other words, it is the
Lebesgue-Stieltjes integral $$int_I ho(gamma(t)),d{mathrm{length_gamma(t), where$${mathrm{length_gamma(t) is the length of the restriction of $$gammato $${sin I:sle t}.Also set

:$$L_ ho(Gamma):=inf_{gammainGamma}L_ ho(gamma).

The area of $$ho is defined as:$$A( ho):=int_D ho^2,dx,dy,and the extremal length of $$Gamma is

:$$EL(Gamma):= sup_ ho frac{L_ ho(Gamma)^2}{A( ho)},,

where the supremum is over all Borel-measureable $$ho:D o [0,infty] with $$0. If $$Gamma contains some non-rectifiable curves and$$Gamma_0 denotes the set of rectifiable curves in $$Gamma, then$$EL(Gamma) is defined to be $$EL(Gamma_0).

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

The extremal distance in $$D between two sets in $$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.

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$$R=(0,w) imes(0,h). Let $$Gamma be the set of all finitelength curves $$gamma:(0,1) o R that cross the rectangle left to right,in the sense that $$lim_{t o 0}gamma(t)is on the left edge $${0} imes [0,h] of the rectangle, and$$lim_{t o 1}gamma(t) is on the right edge $${1} imes [0,h] .(The limits necessarily exist, because we are assuming that $$gammahas finite length.) We will now prove that in this case:$$EL(Gamma)=w/h

First, we may take $$ho=1 on $$R. This $$hogives $$A( ho)=w,h and $$L_ ho(Gamma)=w. The definitionof $$EL(Gamma) as a supremum then gives $$EL(Gamma)ge w/h.

The opposite inequality is not quite so easy. Consider an arbitraryBorel-measurable $$ho:R o [0,infty] such that$$ell:=L_ ho(Gamma)>0.For $$yin(0,h), let $$gamma_y(t)=i,y+w,t(where we are identifying $$R^2 with the complex plane).Then $$gamma_yinGamma, and hence $$ellle L_ ho(gamma_y).The latter inequality may be written as:$$ellle int_0^1 ho(i,y+w,t),w,dt .Integrating this inequality over $$yin(0,h) implies:$$h,ellle int_0^hint_0^1 ho(i,y+w,t),w,dt,dy.Now a change of variable $$x=w,t and an application of the Cauchy-Schwarz inequality give:$$h,ell le int_0^hint_0^w ho(x+i,y),dx,dy le Bigl(int_R ho^2,dx,dyint_R,dx,dyBigr)^{1/2} = igl(w,h,A( ho)igr)^{1/2}. This gives $$ell^2/A( ho)le w/h. Therefore, $$EL(Gamma)le w/h, as required.

As the proof shows, the extremal length of $$Gamma is the same as the extremallength of the much smaller collection of curves $${gamma_y:yin(0,h)}.

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, $$EL(Gamma),EL(Gamma,')=1.It is natural to refer to this as a duality property of extremal length, and a similar duality propertyoccurs in the context of the next subsection. Observe that obtaining a lower bound on$$EL(Gamma) is generally easier than obtaining an upper bound, since the lower bound involveschoosing a reasonably good $$ho and estimating $$L_ ho(Gamma)^2/A( ho),while the upper bound involves proving a statement about all possible $$ho. For this reason,duality is often useful when it can be established: when we know that $$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 theannulus $$A:={zinmathbb C:r_1<|z| and let$$C_1 and $$C_2 be the two boundary componentsof $$A: $$C_1:={z:|z|=r_1}and $$C_2:={z:|z|=r_2}. Consider the extremal distancein $$A between $$C_1 and $$C_2;which is the extremal length of the collection $$Gamma of curves $$gammasubset A connecting $$C_1and $$C_2.

To obtain an lower bound on $$EL(Gamma),we take $$ho(z)=1/|z|. Then for $$gammainGammaoriented from $$C_1 to $$C_2:$$int_gamma |z|^{-1},ds ge int_gamma |z|^{-1},d|z| = int_gamma dlog |z|=log(r_2/r_1).On the other hand,:$$A( ho)=int_A |z|^{-2},dx,dy= int_{0}^{2pi}int_{r_1}^{r_2} r^{-2},r,dr,d heta = 2,pi ,log(r_2/r_1).We conclude that :$$EL(Gamma)ge frac{log(r_2/r_1)}{2pi}.

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 $$ho such that $$ell:=L_ ho(Gamma)>0. For $$hetain [0,2,pi) let $$gamma_ heta:(r_1,r_2) o A denote the curve $$gamma_ heta(r)=e^{i heta}r. Then:$$ellleint_{gamma_ heta} ho,ds =int_{r_1}^{r_2} ho(e^{i heta}r),dr.We integrate over $$heta and apply the Cauchy-Schwarz inequality, to obtain::$$2,pi,ell le int_A ho,dr,d heta le Bigl(int_A ho^2,r,dr,d heta Bigr)^{1/2}Bigl(int_0^{2pi}int_{r_1}^{r_2} frac 1 r,dr,d hetaBigr)^{1/2}.Squaring gives:$$4,pi^2,ell^2le A( ho)cdot,2,pi,log(r_2/r_1).This implies the upper bound $$EL(Gamma)le (2,pi)^{-1},log(r_2/r_1).When combined with the lower bound, this yields the exact value of the extremal length::$$EL(Gamma)=frac{log(r_2/r_1)}{2pi}.

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{2pi}{log(r_2/r_1)}=EL(Gamma)^{-1}.This illustrates another instance of extremal length duality.

Extremal length of topologically essential paths in projective plane

In the above examples, the extremal $$ho which maximized the ratio $$L_ ho(Gamma)^2/A( ho) 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 $$ho, the resulting metric is flat. In the case of the rectangle, this was just the original metric, but for the annulus, the extremal metric identified is the metric of a cylinder. We now discuss an example where an extremal metric is not flat. The projective plane with the spherical metric is obtained by identifying antipodal points on the unit sphere in $$R^3 with its Riemannian spherical metric. In other words, this is the quotient of the sphere by the map $$xmapsto -x. Let $$Gamma denote the set of closed curves in this projective plane that are not null-homotopic. (Each curve in $$Gamma is obtained by projecting a curve on the sphere from a point to its antipode.) Then the spherical metric is extremal for this curve family [Ahlfors (1973)] . (The definition of extremal length readily extends to Riemannian surfaces.) Thus, the extremal length is $$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 $$EL(Gamma)=infty. This follows, for example, by taking :$$ho(z):= egin{cases}(-|z-z_0|,log |z-z_0|)^{-1} & |z-z_0|<1/2,\0 & |z-z_0|ge 1/2,end{cases}which satisfies $$A( ho) and $$L_ ho(gamma)=infty for every rectifiable $$gammainGamma.

Elementary properties of extremal length

The extremal length satisfies a few simple monotonicity properties. First, it is clear that if $$Gamma_1subsetGamma_2, then $$EL(Gamma_1)ge EL(Gamma_2).Moreover, the same conclusion holds if every curve $$gamma_1inGamma_1 contains a curve $$gamma_2in 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:$$EL(Gamma_1cupGamma_2)ge igl(EL(Gamma_1)^{-1}+EL(Gamma_2)^{-1}igr)^{-1}.This is clear if $$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 $$Gamma_1cupGamma_2 are all rectifiable. Let $$ho_1, ho_2 satisfy $$L_{ ho_j}(Gamma_j)ge 1 for $$j=1,2. Set $$ho=max{ ho_1, ho_2}. Then $$L_ ho(Gamma_1cupGamma_2)ge 1 and $$A( ho)=int ho^2,dx,dyleint( ho_1^2+ ho_2^2),dx,dy=A( ho_1)+A( ho_2), which proves the inequality.

Conformal invariance of extremal length

Let $$f:D o D^* be a conformal homeomorphism(a bijective holomorphic map) between planar domains. Suppose that$$Gamma is a collection of curves in $$D,and let $$Gamma^*:={fcirc gamma:gammainGamma} denote theimage curves under $$f. Then $$EL(Gamma)=EL(Gamma^*).This conformal invariance statement is the primary reason why the concept ofextremal length is useful.

Here is a proof of conformal invariance. Let $$Gamma_0 denote the set of curves $$gammainGamma such that $$fcirc gamma is rectifiable, and let$$Gamma_0^*={fcircgamma:gammainGamma_0}, which is the set of rectifiablecurves in $$Gamma^*. Suppose that $$ho^*:D^* o [0,infty] is Borel-measurable. Define:$$ho(z)=|f,'(z)|, ho^*igl(f(z)igr).A change of variables $$w=f(z) gives:$$A( ho)=int_D ho(z)^2,dz,dar z=int_D ho^*(f(z))^2,|f,'(z)|^2,dz,dar z = int_{D^*} ho^*(w)^2,dw,dar w=A( ho^*).Now suppose that $$gammain Gamma_0 is rectifiable, and set $$gamma^*:=fcircgamma. Formally, we may use a change of variables again::$$L_ ho(gamma)=int_gamma ho^*igl(f(z)igr),|f,'(z)|,|dz| = int_{gamma^*} ho(w),|dw|=L_{ ho^*}(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 $$Icap(-infty,t] ,and let $$ell^*(t) be similarly defined with $$gamma^* in place of $$gamma. Then it is easy to see that $$dell^*(t)=|f,'(gamma(t))|,dell(t), and this implies $$L_ ho(gamma)=L_{ ho^*}(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 $$Gamma^* was rectifiable, this wouldprove $$EL(Gamma)=EL(Gamma^*) since we may also apply the above with $$f replaced by its inverseand $$Gamma interchanged with $$Gamma^*. It remains to handle the non-rectifiable curves.

Now let $$hatGamma denote the set of rectifiable curves $$gammainGamma such that $$fcircgamma isnon-rectifiable. We claim that $$EL(hatGamma)=infty.Indeed, take $$ho(z)=|f,'(z)|,h(|f(z)|), where $$h(r)=igl(r,log (r+2)igr)^{-1}.Then a change of variable as above gives:$$A( ho)= int_{D^*} h(|w|)^2,dw,dar w le int_0^{2pi}int_0^infty (r,log (r+2))^{-2} ,r,dr,d hetaFor $$gammainhatGamma and $$rin(0,infty) such that $$fcirc gammais contained in $\{$z:|z|, we have :$$L_ ho(gamma)geinf{h(s):sin [0,r] },mathrm{length}(fcircgamma)=infty.dubiousOn the other hand, suppose that $$gammainhatGamma is such that $$fcircgamma is unbounded.Set $$H(t):=int_0^t h(s),ds. Then$$L_ ho(gamma) is at least the length of the curve $$tmapsto H(|fcirc gamma(t)|)(from an interval in $$R to $$R). Since $$lim_{t oinfty}H(t)=infty,it follows that $$L_ ho(gamma)=infty.Thus, indeed, $$EL(hatGamma)=infty.

Using the results of the previous section, we have:$$EL(Gamma)=EL(Gamma_0cuphatGamma)ge EL(Gamma_0).We have already seen that $$EL(Gamma_0)ge EL(Gamma^*). Thus, $$EL(Gamma)ge EL(Gamma^*).The reverse inequality holds by symmetry, and conformal invariance is therefore established.

ome applications of extremal length

By the calculation of the extremal distance in an annulus and the conformalinvariance it follows that the annulus $${z:r<|z| (where $$0le r)is not conformally homeomorphic to the annulus $\{$w:r^*<|w| if $$frac Rr e frac{R^*}{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. Expand-section|date=June 2008

Discrete extremal length

Suppose that $$G=(V,E) is some graph and $$Gamma is a collection of paths in $$G. There are two variants of extremal length in this setting. To define the edge extremal length, originally introduced by R. J. Duffin [Duffin 1962] , consider a function $$ho:E o [0,infty). The $$ho-length of a path is defined as the sum of $$ho(e) over all edges in the path, counted with multiplicity. The "area" $$A( ho) is defined as $$sum_{ein E} ho(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 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.

Another notion of discrete extremal length that is appropriate in other contexts is vertex extremal length, where $$ho:V o [0,infty), the area is $$A( ho):=sum_{vin V} ho(v)^2, and the length of a path is the sum of $$ho(v) over the vertices visited by the path, with multiplicity.

Notes

References

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