It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces.
Thus, the Schwarzian derivative precisely measures the degree to which a function fails to be a Möbius transformation.
William Thurston interprets the Schwarzian derivative as a measure of how much a conformal map deviates from a Möbius transformation.
Alternatively, consider the second-order linear ordinary differential equation in the complex plane[4] Let
The converse is also true: if such a g exists, and it is holomorphic on a simply connected domain, then two solutions
[6] In particular a sufficient condition for univalence is[7] The Schwarzian derivative and associated second-order ordinary differential equation can be used to determine the Riemann mapping between the upper half-plane or unit circle and any bounded polygon in the complex plane, the edges of which are circular arcs or straight lines.
The accessory parameters that arise as constants of integration are related to the eigenvalues of the second-order differential equation.
Already in 1890 Felix Klein had studied the case of quadrilaterals in terms of the Lamé differential equation.
By the Schwarz reflection principle p(x) extends to a rational function on the complex plane with a double pole at ai: The real numbers βi are called accessory parameters.
Universal Teichmüller space is defined to be the space of real analytic quasiconformal mappings of the unit disc D, or equivalently the upper half-plane H, onto itself, with two mappings considered to be equivalent if on the boundary one is obtained from the other by composition with a Möbius transformation.
Identifying the upper hemisphere with D, Lipman Bers used the Schwarzian derivative to define a mapping which embeds universal Teichmüller space into an open subset U of the space of bounded holomorphic functions g on D with the uniform norm.
Frederick Gehring showed in 1977 that U is the interior of the closed subset of Schwarzian derivatives of univalent functions.
[11][12][13] For a compact Riemann surface S of genus greater than 1, its universal covering space is the unit disc D on which its fundamental group Γ acts by Möbius transformations.
The holomorphic functions g have the property that is invariant under Γ, so determine quadratic differentials on S. In this way, the Teichmüller space of S is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on S. The transformation property allows the Schwarzian derivative to be interpreted as a continuous 1-cocycle or crossed homomorphism of the diffeomorphism group of the circle with coefficients in the module of densities of degree 2 on the circle.
onto G is the identity map; the correspondence is by the map The crossed homomorphisms form a vector space and containing as a subspace the coboundary crossed homomorphisms in M. A simple averaging argument shows that, if K is a compact group and V a topological vector space on which K acts continuously, then the higher cohomology groups vanish In particular for 1-cocycles χ with averaging over y, using left invariant of the Haar measure on K gives with Thus by averaging it may be assumed that c satisfies the normalisation condition
By continuity this reduces to the computation of crossed homomorphisms 𝜙 of the Witt algebra into Fλ(S1).
From this condition and the recurrence relation, it follows that up to scalar multiples, this has a unique non-zero solution when
Taking boundary values, universal Teichmüller can be identified with the quotient of the group of quasisymmetric homeomorphisms QS(S1) by the subgroup of Möbius transformations Moeb(S1).
The inverse of the diffeomorphism f sends the Hill's operator to The Schwarzian derivative and the other 1-cocycle defined on Diff(S1) can be extended to biholomorphic between open sets in the complex plane.
Conversely if Γ is flat it is uniquely determined by A: a biholomorphism f on U is contained in Γ in if and only if the power series of T–f(a) ∘ f ∘ Ta lies in A for every a in U: in other words the formal power series for f at a is given by an element of A with z replaced by z − a; or more briefly all the jets of f lie in A.
This group acts faithfully on the space of polynomials of degree k (truncating terms of order higher than k).
A flat pseudogroup Γ is said to be "defined by differential equations" if there is a finite integer k such that homomorphism of A into Gk is faithful and the image is a closed subgroup.
It contains the polynomial vectors fields with basis dn = zn+1 d/dz (n ≥ 0), which is a subalgebra of the Witt algebra.
Again these act on the space of polynomials of degree ≤ k by differentiation—it can be identified with C[[z]]/(zk+1)—and the images of d0, ..., dk – 1 give a basis of the Lie algebra of Gk.
If the 1-cocycle 𝜙 satisfies suitable continuity or analyticity conditions, it induces a 1-cocycle of holomorphic vector fields, also compatible with restriction.
Accordingly, it defines a 1-cocycle on holomorphic vector fields on C:[18] Restricting to the Lie algebra of polynomial vector fields with basis dn = zn+1 d/dz (n ≥ −1), these can be determined using the same methods of Lie algebra cohomology (as in the previous section on crossed homomorphisms).
There the calculation was for the whole Witt algebra acting on densities of order k, whereas here it is just for a subalgebra acting on holomorphic (or polynomial) differentials of order k. Again, assuming that 𝜙 vanishes on rotations of C, there are non-zero 1-cocycles, unique up to scalar multiples.
More significantly they can be used to define corresponding affine or projective structures and connections on Riemann surfaces.
[19] Gunning in 1966 describes how this process can be reversed: for genus p > 1, the existence of a projective connection, defined using the Schwarzian derivative 𝜙2 and proved using standard results on cohomology, can be used to identify the universal covering surface with the upper half plane or unit disk (a similar result holds for genus 1, using affine connections and 𝜙1).
Another generalization applies to positive curves in a Lagrangian Grassmannian (Ovsienko & Tabachnikov 2005).