Liouville's theorem (conformal mappings)

In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that any smooth conformal mapping on a domain of Rn, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations (in n dimensions). This theorem severely limits the variety of possible conformal mappings in R3 and higher-dimensional spaces. By contrast, conformal mappings in R2 can be much more complicated – for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem. Generalizations of the theorem hold for transformations that are only weakly differentiable . The focus of such a study is the non-linear Cauchy–Riemann system that is a necessary and sufficient condition for a smooth mapping f : Ω → Rn to be conformal: where Df is the Jacobian derivative, T is the matrix transpose, and I is the identity matrix. A weak solution of this system is defined to be an element f of the Sobolev space W(Ω, Rn) with non-negative Jacobian determinant almost everywhere, such that the Cauchy–Riemann system holds at almost every point of Ω. Liouville's theorem is then that every weak solution (in this sense) is a Möbius transformation, meaning that it has the form where a, b are vectors in Rn, α is a scalar, A is a rotation matrix, ε = 0 or 2, and the matrix in parentheses is I or a Householder matrix (so, orthogonal). Equivalently stated, any quasiconformal map of a domain in Euclidean space that is also conformal is a Möbius transformation. This equivalent statement justifies using the Sobolev space W1,n, since f ∈ W(Ω, Rn) then follows from the geometrical condition of conformality and the ACL characterization of Sobolev space. The result is not optimal however: in even dimensions n = 2k, the theorem also holds for solutions that are only assumed to be in the space W, and this result is sharp in the sense that there are weak solutions of the Cauchy–Riemann system in W1,p for any p < k that are not Möbius transformations.