Conformal maps in higher dimensions - MathOverflow

Conformal maps in higher dimensions

2009-12-29T21:09:44Z

In dimension 2 we know by the Riemann mapping theorem that any simply connected domain ( $\neq \mathbb{R}^{2}$) can be mapped bijectively to the unit disk with a function that preserves angles between curves, ie is conformal.

I have read the claim that conformal maps in higher dimensions are pretty boring but does anyone know a proof or even a intuitive argument that conformal maps in higher dimensions are trivial?

Answer by Andy Putman for Conformal maps in higher dimensions

2009-12-29T21:19:25Z

I think you're looking for Liouville's theorem. This theorem states that for $n >2$, if $V_1,V_2 \subset \mathbb{R}^n$ are open subsets and $f : V_1 \rightarrow V_2$ is a smooth conformal map, then $f$ is the restriction of a higher-dimensional analogue of a Mobius transformation.

By the way, observe that there are no assumptions on the topology of the $V_i$ -- they don't have to be simply-connected, etc.

Answer by Gjergji Zaimi for Conformal maps in higher dimensions

2009-12-29T21:38:28Z

Because the Riemann mapping theorem does not hold in higher dimensions. While there are all sorts of conformal mappings in dimension 2, for higher dimensions Liouville's theorem restricts all possible conformal mappings to the ones that are compositions of similarities, translations, orthogonal transformations and inversions. In generality there are contractible spaces not homeomorphic (therefore not conformal) to $\mathbb{R}^n$ such as Whitehead continuum

As for the proof of Liouville's theorem, maybe this article is of interest where you can see a sketch of Nevanlinna's original proof and a proof by nonstandard analysis.

Answer by Benoît Kloeckner for Conformal maps in higher dimensions

2011-09-03T06:57:09Z

There is a proof of Liouville's theorem by Charles Frances with less computations than most others, and carrying some intuition. However it is restricted to real-analytic transforms, and it is written in French. It has been published in "L'enseignement mathématique", and is available there: http://www.math.u-psud.fr/~frances/liouville2.pdf

Answer by Changyu Guo for Conformal maps in higher dimensions

2012-09-28T11:39:47Z

I think this is a good reference for it. Iwaniec, Tadeusz; Martin, Gaven Geometric function theory and non-linear analysis. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2001. xvi+552.