Extremal length

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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

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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

 

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

 

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 (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

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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

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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

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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  

 

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

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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

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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

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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

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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

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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

 

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

 .[dubiousdiscuss]

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

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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

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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

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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

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  1. ^ Ahlfors (1973)
  2. ^ Duffin 1962

References

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  • 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