Schwarzian derivative

In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all linear fractional transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces.

Definition

The Schwarzian derivative of a holomorphic function f of one complex variable z is defined by

$(Sf)(z) = \left( \frac{f''(z)}{f'(z)}\right)' - \frac{1}{2}\left({f''(z)\over f'(z)}\right)^2 = \frac{f'''(z)}{f'(z)}-\frac{3}{2}\left({f''(z)\over f'(z)}\right)^2.$

The same formula also defines the Schwarzian derivative of a C³ function of one real variable. The alternative notation

$\{f,z\} = (Sf)(z)$

is frequently used.

Properties

The Schwarzian derivative of any fractional linear transformation

$g(z) = \frac{az + b}{cz + d}$

is zero. Conversely, the fractional linear transformations are the only functions with this property. Thus, the Schwarzian derivative precisely measures the degree to which a function fails to be a fractional linear transformation.

If g is a fractional linear transformation, then the composition g o f has the same Schwarzian derivative as f. On the other hand, the Schwarzian derivative of f o g is given by the chain rule

$(S(f \circ g))(z) = (Sf)(g(z)) \cdot g'(z)^2.$

More generally, for any sufficiently differentiable functions f and g

$S(f \circ g) = \left( S(f)\circ g\right ) \cdot(g')^2+S(g).$

This makes the Schwarzian derivative an important tool in one-dimensional dynamics since it implies that all iterates of a function with negative Schwarzian will also have negative Schwarzian.

Introducing the function of two complex variables

$F(z,w)= \log \left ( \frac{f(z)-f(w)}{z-w} \right ),$

its second mixed partial derivative is given by

$\frac{\partial^2 F(z,w)}{\partial z \, \partial w} = {f^\prime(z)f^\prime(w)\over(f(z)-f(w))^2}-{1\over(z-w)^2},$

and the Schwarzian derivative is given by the formula:

$(Sf)(z)= \left. 6 \cdot {\partial^2 F(z,w)\over \partial z \partial w}\right\vert_{z=w}.$

The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has

$(Sw)(v) = -\left(\frac{dw}{dv}\right)^2 (Sv)(w)$

which follows from the inverse function theorem, namely that $v'(w)=1/w'.$

Differential equation

The Schwarzian derivative has a fundamental relation with a second-order linear ordinary differential equation in the complex plane. Let $f_1(z)$ and $f_2(z)$ be two linearly independent holomorphic solutions of

$\frac{d^2f}{dz^2}+ Q(z) f(z)=0.$

Then the ratio $g(z)=f_1(z)/f_2(z)$ satisfies

$(Sg)(z) = 2Q(z)$

over the domain on which $f_1(z)$ and $f_2(z)$ are defined, and $f_2(z) \ne 0.$ The converse is also true: if such a g exists, and it is holomorphic on a simply connected domain, then two solutions $f_1$ and $f_2$ can be found, and furthermore, these are unique up to a common scale factor.

When a linear second-order ordinary differential equation can be brought into the above form, the resulting Q is sometimes called the Q-value of the equation.

Note that the Gaussian hypergeometric differential equation can be brought into the above form, and thus pairs of solutions to the hypergeometric equation are related in this way.

Conditions for univalence

If f is a holomorphic function on the unit disc, D, then W. Kraus (1932) and Nehari (1949) proved that a necessary condition for f to be univalent is

$|S(f)| \le 6(1-|z|^2)^{-2}.$

Conversely if f(z) is a holomorphic function on D satisfying

$|S(f)(z)| \le 2(1-|z|^2)^{-2},$

then Nehari proved that f is univalent.

In particular a sufficient condition for univalence is

$|S(f)|\le 2.$

Conformal mapping of circular arc polygons

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. For polygons with straight edges, this reduces to the Schwarz–Christoffel mapping, which can be derived directly without using the Schwarzian derivative. 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.

Let Δ be a circular arc polygon with angles πα₁, ..., πα_n in clockwise order. Let f : H → Δ be a holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points a₁, ..., a_n on the real axis. Then p(x) = S(f)(x) is real-valued for x real and not one of the points. By the Schwarz reflection principle p(x) extends to a rational function on the complex plane with a double pole at a_i:

$p(z)=\sum_{i=1}^n \frac{(1-\alpha_i^2)}{2(z-a_i)^2} + \frac{\beta_i}{z-a_i}.$

The real numbers β_i are called accessory parameters. They are subject to 3 linear constraints:

$\sum \beta_i=0$

$\sum 2a_i \beta_i + \left ( 1-\alpha_i^2 \right ) =0$

$\sum a_i^2 \beta_i + a_i \left ( 1-\alpha_i^2 \right ) =0$

which correspond to the vanishing of the coefficients of $z^{-1}, z^{-2}$ and $z^{-3}$ in the expansion of p(z) around z = ∞. The mapping f(z) can then be written as

$f(z) = {u_1(z)\over u_2(z)},$

where $u_1(z)$ and $u_2(z)$ are linearly independent holomorphic solutions of the linear second order ordinary differential equation

$u^{\prime\prime}(z) + \tfrac{1}{2} p(z)u(z)=0.$

There are n−3 linearly independent accessory parameters, which can be difficult to determine in practise.

For a triangle, when n = 3, there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and f(z) can be written in terms of hypergeometric functions.

For a quadrilateral the accessory parameters depend on one independent variable λ. Writing U(z) = q(z)u(z) for a suitable choice of q(z), the ordinary differential equation takes the form

$a(z) U^{\prime\prime}(z) + b(z) U^\prime(z) +(c(z)+\lambda)U(z)=0.$

Thus $q(z) u_i(z)$ are eigenfunctions of a Sturm-Liouville equation on the interval $[a_i,a_{i+1}]$. By the Sturm separation theorem, the non-vanishing of $u_2(z)$ forces λ to be the lowest eigenvalue.

Complex structure on Teichmüller space

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 D with the lower hemisphere of the Riemann sphere, any quasiconformal self-map f of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere $\tilde{f}$ onto itself. In fact $\tilde{f}$ is determined as the restriction to the upper hemisphere of the solution of the Beltrami differential equation

$\frac{\partial F}{\partial \overline{z}} = \mu(z) \frac{\partial F}{\partial z},$

where μ is the bounded measurable function defined by

$\mu(z) = {{\partial f\over \partial \overline{z}}\over{\partial f\over \partial z}}$

on the lower hemisphere, extended to 0 on the upper hemisphere.

Identifying the upper hemisphere with D, Lipman Bers used the Schwarzian derivative to define a mapping

$g= S(\tilde{f}),$

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.

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 Teichmüller space of S can be identified with the subspace of the universal Teichmüller space invariant under Γ. The holomorphic functions g have the property that

$g(z) dz^{2}$

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.

Diffeomorphism group of the circle

Let F_λ(S¹) be the space of tensor densities of degree λ on S¹. The group of orientation-preserving diffeomorphisms of S¹, Diff(S¹), acts on F_λ(S¹) via pushforwards. If f is an element of Diff(S¹) then consider the mapping

$f \to S(f^{-1}).$

In the language of group cohomology the chain-like rule above says that this mapping is a 1-cocycle on Diff(S¹)with coefficients in F₂(S¹). In fact

$H^1(\text{Diff}(\mathbf{S}^1);F_2) = \mathbf{R}$

and the 1-cocycle generating the cohomology is f → S(f⁻¹).

There is an infinitesimal version of this result giving a 1-cocycle for the Lie algebra Vect(S¹) of vector fields. This in turn gives the unique non-trivial central extension of Vect(S¹), the Virasoro algebra.

The group Diff(S¹) and its central extension also appear naturally in the context of Teichmüller theory and string theory. In fact the homeomorphisms of S¹ induced by quasiconformal self-maps of D are precisely the quasisymmetric homeomorphisms of S¹; these are exactly homeomorphisms which do not send four points with cross ratio 1/2 to points with cross ratio near 1 or 0. 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). (It can also be realized naturally as the space of quasicircles in C.) Since

$\text{Moeb}(\mathbf{S}^1)\subset \text{Diff}(\mathbf{S}^1) \subset \text{QS}(\mathbf{S}^1)$

the homogeneous space Diff(S¹)/Moeb(S¹) is naturally a subspace of universal Teichmüller space. It is also naturally a complex manifold and this and other natural geometric structures are compatible with those on Teichmüller space. The dual of the Lie algebra of Diff(S¹) can be identified with the space of Hill's operators on S¹

${d^2\over d\theta^2} + q(\theta),$

and the coadjoint action of Diff(S¹) invokes the Schwarzian derivative. The inverse of the diffeomorphism f sends the Hill's operator to

${d^2\over d\theta^2} + f^\prime(\theta)^2 \,q\circ f(\theta) + \tfrac{1}{2} S(f)(\theta).$

References

• , Chapter 6, "Teichmüller Spaces"
• , Chapter 10, "The Schwarzian".
• , Section 12, "Mapping of polygons with circular arcs".
• , "On the theory of generalized Lamé functions".