In this post, I wanted to share a neat proof that conformally isomorphic annuli have common radial ratio. There are a number of ways to go about this, but I’ve always found this technique particularly striking, as it relates to the broader theory of conformal invariants and extremal length.
Suppose we have a biholomorphic map , where . By inverting if necessary, we can assume maps the inner circle to the inner circle and maps the outer circle to the outer circle. Expand as a Laurent series in the annulus, say
The area bounded by a Jordan curve is Then
We can plug in Laurent series for and that of its derivative to get
This holds for , but by continuity, it also holds for and . If we set , we get If we set , we get We can combine these to find that
Regardless of whether is positive, negative, or zero, each of the terms is nonnegative!
This shows that . Applying the same argument with replaced by shows that , so