Ahlfors' proof of locally K-quasiconformal to K-quasiconformal

Question

This is a question I originally posted in Math Stack Exchange, but perhaps the question was too specialized, so I thought I'd post it here instead

I'm currently reading through "Lectures on quasiconformal mappings." I'm a bit confused about the proof of Theorem 1 in Chapter 2.

Here's the statement of the theorem and the proof.

I'm actually confused about the Editors' note on the proof of this theorem. It says:

"Shishikura has pointed out to us that the existence of a 'sufficiently fine' subdivision requires proof...

First subdivide Q by both vertical and horizontal lines so that each small rectangle has modulus less than 1/K and any pair of vertically adjacent small rectangles has a neighborhood in which $f$ is K-q.c. The image of each small rectangle then has modulus less than 1, so one can show by using the Teichmüller extremal problem in Chapter III A that it contains a horizontal line segment...."

I don't understand how the Teichmüller extremal problem is related to finding the horizontal segments and I don't understand why having the image of the small rectangles have module less than 1 allows one to find horizontal segments.

Answer 1

I can perhaps imagine roughly what the connection to Teichmüller's extremal problem would be, but I'm having enough difficulty unpacking both Ahlfor's proof and Shishikura's remark that I don't think it's worth the effort.

I personally find Ahlfor's book too out-of-date and too sparse with the details. Instead, I would recommend Väisälä's book Lectures on $n$-dimensional quasiconformal mappings, which is very carefully written and which presents the classical theory of quasiconformal mappings in essentially the final form. One difference is that "module of a quadrilateral" is replaced with the notion of modulus of an arbitrary family of curves, which is technically much nicer to work with. For the local-to-global proof, one would show that the geometric definition (quasi-invariance of modulus) is equivalent to the analytic definition, that the mapping $f$ is in $W_{\text{loc}}^{1,n}(\Omega, \mathbb{R}^n)$ and satisfies $|Df(x)|^n \leq KJ_f(x)$ for a.e. $x$. But for the analytic definition, the local-to-global property is trivial.

Answer 2

If you are still interested in the this problem and in how to make this machinery workS, my suggestion is to taking a carefully reading in the notes by Chris Bishop available in

https://www.math.stonybrook.edu/~bishop/classes/math627.S18/

The "figure" present in Ahlfors' book is essentially the correct idea but to turn the picture in a proof take some efforts. The proof by Bishop is quite clean and easy to follows; I will not try to reproduce it here.