Liouville’s theorem states
that in Euclidean space of dimension greater than two, every conformal mapping
must, by necessity, be an elementary transformation (i.e., a translation, a
magnification, an orthogonal transformation, a reflection through reciprocal
radii, or a combination of these transformations). This theorem was proven
by R. Nevanlinna under the additional assumption that the mappings be
at least four times differentiable. We show that a modified version of
Nevanlinna’s proof is still valid when the mappings are assumed to be only twice
differentiable. Our methods are those of Nonstandard Analysis as developed by A.
Robinson.
