Extremal Length - Conformal Invariance of Extremal Length

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

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

.

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.