Möbius characterization by the Schwarzian

Statement

Let $D\subseteq\mathbb C$ be a connected domain, and let $f:D\to\widehat{\mathbb C}$ be locally univalent and meromorphic. Then

S(f)=0

if and only if $f$ is the restriction to $D$ of a Möbius transformation.

Proof idea

Every Möbius transformation has zero Schwarzian by direct substitution. Conversely, the differential equation $S(f)=0$ has local solutions

f(z)=\frac{az+b}{cz+d}, \qquad ad-bc\ne0.

These local Möbius transformations agree on overlaps because they agree with $f$ . Connectedness and the identity theorem therefore produce one Möbius transformation on all of $D$ .

Relation to equal Schwarzians

Applying this result after passing to a local inverse yields the theorem that two locally univalent maps with equal Schwarzians differ by Möbius postcomposition.

References

Zeev Nehari, Conformal Mapping, Dover, 1975. Relevant: Chapter VI.