5
$\begingroup$

A mapping $T: \Bbb R^n\to \Bbb R^n$ is said to be conformal if it is bijective and preserves angles, i.e., if $x, y: [0,1]\to \Bbb R^n$ are curves with $x(t_0)=y(t_0)$ then

$$\cos (Tx(t_0),Ty(t_0))= \cos (x(t_0),y(t_0))= \frac{x'(t_0)\cdot y'(t_0)}{|x'(t_0)|| y'(t_0)|}.$$

A typical example of conformal mapping is the inversion $I:\Bbb R^n \to \Bbb R^n$ $I(x)= \frac{x}{|x|^2}$, with the convention that $I(0)=\infty $ and $I(\infty)=0$.

Trivial examples are rigid motions i.e., a combination of orthogonal group, Scallings or homothety and/or translations.

I am barely looking for a proof or a reference for the following Theorem:

Theorem: Every conformal mapping is the composite of finely many rigid motions and the Inversion mapping.

There are several book complex analysis dealing with the case $n=2$ on the complex plane.

But I haven't seen any for the higher dimensional situation.

$\endgroup$
1

4 Answers 4

7
$\begingroup$

My two cents: a proof for $n=3$ is given explicitly by Dubrovin, Fomenko and Novikov in [1], §15.2 pp. 138-142. The authors explain also how to extend their proof to the case $n>3$ and leave the details as an exercise.

An addendum

The answer by @Piotr Hajlasz triggered my curiosity and pushed me to go a little bit beyond reference [1], which requires a $C^4$ regularity on the conformal map considered ([1], §15.2 p. 138).
According to Caraman ([2] section 3, chapter 2, p. 358), the proof of Liouville's theorem requiring a minimal regularity on the mapping was given first by Reshetnyak in [3]. Reshetnyak assumes the mapping to be of class $W^1_n$: while the paper is short, the offered proof is highly non trivial.

Reference

[1] Boris A. Dubrovin, Analtoly T. Fomenko and Sergey P. Novikov, Modern geometry - methods and applications. Part I. The geometry of surfaces, transformation groups, and fields, translated by Robert G. Burns. 2nd ed. (English) Graduate Texts in Mathematics, 93, Berlin-Heidelberg-New York: Springer-Verlag, pp. xv+468 (1992), MR1138462, Zbl 0751.53001

[2] Petru Caraman, $n$-Dimensional Quasiconformal (QCF) Mappings, revised, enlarged and translated from the Romanian by the Author (English), Tunbridge Wells, Kent: Abacus Press, pp. 551 (1974), ISBN 0-85626-005-3, MR0357782, Zbl 0342.30015.

[3] Yuriĭ G. Reshetnyak, "Liouville’s theorem on conformal mappings for minimal regularity assumptions", (English, translated from the Russian), Siberian Mathematical Journal 8 (1967), pp. 631-634 (1968), MR0218544, Zbl 0167.36102.

$\endgroup$
9
$\begingroup$

See the following Wikipedia page: https://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)

$\endgroup$
6
$\begingroup$

Theorem (Liouville). If $\Omega\subset\mathbb{R}^n$, $n\geq 3$ is open and $f:\Omega\to\mathbb{R}^n$ is conformal, then $f$ is a Mobius transformation.

While the theorem is true for $f\in C^1$, there is no easy proof in that case. Standard proofs assume that $f\in C^3$ or even $f\in C^4$. The classical and well know proof due to Nevanlinna can be found here (see page 265):

http://www.pitt.edu/~hajlasz/Notatki/Differential_Geometry_1.pdf

$\endgroup$
1
$\begingroup$

Have you tried the first chapter of Riccardo Benedetti Carlo Petronio, Lectures on Hyperbolic Geometry? It contains a proof of Liouville's theorem from which you can easily deduce the result you are looking for.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.