Liouville's theorem on conformal mappings

Remarks

Proof idea: the condition $Df^T Df = \lambda^2 I$ is a system of partial differential equations for $f$. Differentiating it and using $n \ge 3$ forces the function $1/\lambda$ to be either constant or of the form $a \abs{x - x_0}^2$ with $a > 0$. In the first case $f$ is a similarity; in the second, composing with an inversion centred at $x_0$ reduces to the first case. The hypothesis $n \ge 3$ is necessary: in $\R^2 = \C$, every holomorphic function with nonzero derivative is conformal, for example $z \mapsto e^z$, and it is not a Möbius transformation since it is not injective. The smoothness hypothesis can be weakened a great deal. By results of Gehring and of Reshetnyak, a map of Sobolev class $W^{1,n}_{\mathrm{loc}}$ whose derivative is conformal almost everywhere is constant or a restricted Möbius transformation.

Going deeper

For a student

The theorem explains why conformal geometry in dimension $n \ge 3$ is rigid: the conformal maps form the finite-dimensional group of Möbius transformations, of dimension $(n+1)(n+2)/2$. In dimension $2$, by contrast, the Riemann mapping theorem gives a conformal map between any two simply connected proper open subsets of the plane. The result is the local form of the fact that the conformal group of the round sphere $S^n$, $n \ge 3$, is the Möbius group. It is a starting point for the theory of quasiconformal maps in higher dimensions.