Schwarzian derivative

In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius 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. It is named after the German mathematician Hermann Schwarz.

Definition

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

:$(Sf)(z) = \left( \frac\right)' - \frac\left(\right)^2 = \frac-\frac\left(\right)^2.$

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

:$\ = (Sf)(z)$

is frequently used.

Properties

The Schwarzian derivative of any Möbius transformation

: $g(z) = \frac$

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

If is a Möbius transformation, then the composition has the same Schwarzian derivative as ; and on the other hand, the Schwarzian derivative of 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 and

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

When and are smooth real-valued functions, this implies that all iterations of a function with negative (or positive) Schwarzian will remain negative (resp. positive), a fact of use in the study of one-dimensional dynamics.

Introducing the function of two complex variables

:$F(z,w)= \log \left ( \frac \right ),$

its second mixed partial derivative is given by

:$\frac = -,$

and the Schwarzian derivative is given by the formula:

:$(Sf)(w)= \left. 6 \cdot \right\vert_.$

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

:$(Sw)(v) = -\left(\frac\right)^2 (Sv)(w)$

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

:$\frac+ Q(z) f(z)=0.$

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

:$, S(f), \le 6(1-, z, ^2)^.$

Conversely if is a holomorphic function on satisfying

:$, S(f)(z), \le 2(1-, z, ^2)^,$

then Nehari proved that 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 be a circular arc polygon with angles in clockwise order. Let be a holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points on the real axis. Then is real-valued for real and not one of the points. By the Schwarz reflection principle extends to a rational function on the complex plane with a double pole at :

:$p(z)=\sum_^n \frac + \frac.$

The real numbers are called ''accessory parameters''. They are subject to three 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 $z^, z^$ and $z^$ in the expansion of around . The mapping can then be written as

:$f(z) = ,$

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

:$u^(z) + \tfrac p(z)u(z)=0.$

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

For a triangle, when , there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and is the Schwarz triangle function, which can be written in terms of hypergeometric functions.

For a quadrilateral the accessory parameters depend on one independent variable . Writing for a suitable choice of , the ordinary differential equation takes the form

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

Thus $q(z) u_i(z)$ are eigenfunctions of a Sturm–Liouville equation on the interval _i,a_/math>. By the Sturm separation theorem, the non-vanishing of $u_2(z)$ forces 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 , or equivalently the upper half-plane , 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 with the lower hemisphere of the Riemann sphere, any quasiconformal self-map of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere $\tilde$ onto itself. In fact $\tilde$ is determined as the restriction to the upper hemisphere of the solution of the Beltrami differential equation

:$\frac = \mu(z) \frac,$

where μ is the bounded measurable function defined by

:$\mu(z) = \frac \bigg/ \frac$

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

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

:$g= S(\tilde),$

which embeds universal Teichmüller space into an open subset of the space of bounded holomorphic functions on with the uniform norm. Frederick Gehring showed in 1977 that is the interior of the closed subset of Schwarzian derivatives of univalent functions.

For a compact Riemann surface of genus greater than 1, its universal covering space is the unit disc on which its fundamental group acts by Möbius transformations. The Teichmüller space of can be identified with the subspace of the universal Teichmüller space invariant under . The holomorphic functions have the property that

:$g(z) \, dz^2$

is invariant under , so determine quadratic differentials on . In this way, the Teichmüller space of is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on .

Diffeomorphism group of the circle

Crossed homomorphisms

The transformation property

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

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.
Let be the space of tensor densities of degree on . The group of orientation-preserving diffeomorphisms of , acts on via pushforwards. If is an element of then consider the mapping

:$f \to S(f^).$

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

:$H^1(\text(\mathbf^1);F_2 (\mathbf^1)) = \mathbf$

and the 1-cocycle generating the cohomology is . The computation of 1-cohomology is a particular case of the more general result

:$H^1(\text(\mathbf^1);F_\lambda (\mathbf^1)) = \mathbf\,\, \mathrm \,\, \lambda=0,1,2\,\, \mathrm \,\,(0) \,\,\mathrm$

Note that if is a group and a -module, then the identity defining a crossed homomorphism of into can be expressed in terms of standard homomorphisms of groups: it is encoded in a homomorphism of into the semidirect product $M\rtimes G$ such that the composition of with the projection $M\rtimes G$ onto 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 for in . A simple averaging argument shows that, if is a compact group and a topological vector space on which ''K'' acts continuously, then the higher cohomology groups vanish for . In particular for 1-cocycles χ with

:$\chi(xy) = \chi(x) + x\cdot \chi(y),$

averaging over , using left invariant of the Haar measure on gives

:$\chi(x) = m - x\cdot m,$

with

:$m=\int_K \chi(y)\,dy.$

Thus by averaging it may be assumed that satisfies the normalisation condition for in . Note that if any element in satisfies then . But then, since is a homomorphism,
, so that satisfies the equivariance condition . Thus it may be assumed that the cocycle satisfies these normalisation conditions for . The Schwarzian derivative in fact vanishes whenever is a Möbius transformation corresponding to . The other two 1-cycles discussed below vanish only on .

There is an infinitesimal version of this result giving a 1-cocycle for , the Lie algebra of smooth vector fields, and hence for the Witt algebra, the subalgebra of trigonometric polynomial vector fields. Indeed, when is a Lie group and the action of on is smooth, there is a Lie algebraic version of crossed homomorphism obtained by taking the corresponding homomorphisms of the Lie algebras (the derivatives of the homomorphisms at the identity). This also makes sense for and leads to the 1-cocycle

:$s\left(f\, \right) = \,(d\theta)^2$

which satisfies the identity

:s( ,Y=X\cdot s(Y) -Y\cdot s(X).

In the Lie algebra case, the coboundary maps have the form for in . In both cases the 1-cohomology is defined as the space of crossed homomorphisms modulo coboundaries. The natural correspondence between group homomorphisms and Lie algebra homomorphisms leads to the "van Est inclusion map"

:$H^1(\operatorname(\mathbf^1);F_\lambda (\mathbf^1)) \hookrightarrow H^1(\operatorname(\mathbf^1);F_\lambda (\mathbf^1)),$

In this way the calculation can be reduced to that of Lie algebra cohomology. By continuity this reduces to the computation of crossed homomorphisms of the Witt algebra into . The normalisations conditions on the group crossed homomorphism imply the following additional conditions for :

:$\varphi(\operatorname(x) X) = x\cdot \varphi(X),\,\, \varphi(d/d\theta) = 0$

for in .

Following the conventions of , a basis of the Witt algebra is given by

:$d_n = i e^ \,$

so that . A basis for the complexification of is given by

:$v_n=e^ \, (d\theta)^\lambda,$

so that

:$d_m \cdot v_n = -(n+\lambda m)v_,\,\, g_\zeta \cdot v_n = \zeta^ v_n,$

for in . This forces for suitable coefficients . The crossed homomorphism condition
gives a recurrence relation for the :

:$(m-n) a_ = (m+\lambda n) a_m-(n+\lambda m)a_n.$

The condition , implies that . From this condition and the recurrence relation, it follows that up to scalar multiples, this has a unique non-zero solution when equals 0, 1 or 2 and only the zero solution otherwise. The solution for corresponds to the group 1-cocycle $\varphi_1(f) =f^/f^\prime\, d\theta$. The solution for corresponds to the group 1-cocycle . The corresponding Lie algebra 1-cocycles for are given up to a scalar multiple by

:$\varphi_\lambda\left(F \right) = \, (d\theta)^\lambda.$

Central extensions

The crossed homomorphisms in turn give rise to the central extension of and of its Lie algebra , the so-called Virasoro algebra.

Coadjoint action

The group and its central extension also appear naturally in the context of Teichmüller theory and string theory. In fact the homeomorphisms of induced by quasiconformal self-maps of are precisely the quasisymmetric homeomorphisms of ; 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 by the subgroup of Möbius transformations . (It can also be realized naturally as the space of quasicircles in .) Since

:$\operatorname(\mathbf^1)\subset \operatorname(\mathbf^1) \subset \text(\mathbf^1)$

the homogeneous space 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 can be identified with the space of Hill's operators on

:$+ q(\theta),$

and the coadjoint action of invokes the Schwarzian derivative. The inverse of the diffeomorphism sends the Hill's operator to

:$+ f^\prime(\theta)^2 \,q\circ f(\theta) + \tfrac S(f)(\theta).$

Pseudogroups and connections

The Schwarzian derivative and the other 1-cocycle defined on can be extended to biholomorphic between open sets in the complex plane. In this case the local description leads to the theory of analytic pseudogroups, formalizing the theory of infinite-dimensional groups and Lie algebras first studied by Élie Cartan in the 1910s. This is related to affine and projective structures on Riemann surfaces as well as the theory of Schwarzian or projective connections, discussed by Gunning, Schiffer and Hawley.

A holomorphic pseudogroup on consists of a collection of biholomorphisms between open sets and in which contains the identity maps for each open , which is closed under restricting to opens, which is closed under composition (when possible), which is closed under taking inverses and such that if a biholomorphisms is locally in , then it too is in . The pseudogroup is said to be ''transitive'' if, given and in , there is a biholomorphism in such that . A particular case of transitive pseudogroups are those which are ''flat'', i.e. contain all complex translations . Let be the group, under composition, of formal power series transformations with . A holomorphic pseudogroup defines a subgroup of , namely the subgroup defined by the Taylor series expansion about 0 (or "jet") of elements of with . Conversely if is flat it is uniquely determined by : a biholomorphism on is contained in in if and only if the power series of lies in for every in : in other words the formal power series for at is given by an element of with replaced by ; or more briefly all the jets of lie in .

The group has a natural homomorphisms onto the group of -jets obtained by taking the truncated power series taken up to the term ''z''^''k''. This group acts faithfully on the space of polynomials of degree (truncating terms of order higher than ''k''). Truncations similarly define homomorphisms of onto ; the kernel consists of maps ''f'' with , so is Abelian. Thus the group ''G''_''k'' is solvable, a fact also clear from the fact that it is in triangular form for the basis of monomials.

A flat pseudogroup is said to be ''"defined by differential equations"'' if there is a finite integer such that homomorphism of into $''G''$_''k'' is faithful and the image is a closed subgroup. The smallest such is said to be the ''order'' of .
There is a complete classification of all subgroups that arise in this way which satisfy the additional assumptions that the image of in is a complex subgroup and that equals : this implies that the pseudogroup also contains the scaling transformations for , i.e. contains contains every polynomial with .

The only possibilities in this case are that and ; or that and . The former is the pseudogroup defined by affine subgroup of the complex Möbius group (the transformations fixing ); the latter is the pseudogroup defined by the whole complex Möbius group.

This classification can easily be reduced to a Lie algebraic problem since the formal Lie algebra $\mathfrak$ of consists of formal vector fields with ''F'' a formal power series. It contains the polynomial vectors fields with basis , which is a subalgebra of the Witt algebra. The Lie brackets are given by . Again these act on the space of polynomials of degree by differentiation—it can be identified with —and the images of give a basis of the Lie algebra of . Note that . Let $\mathfrak$ denote the Lie algebra of : it is isomorphic to a subalgebra of the Lie algebra of . It contains and is invariant under . Since $\mathfrak$ is a Lie subalgebra of the Witt algebra, the only possibility is that it has basis or basis for some . There are corresponding group elements of the form . Composing this with translations yields with . Unless , this contradicts the form of subgroup ; so .

The Schwarzian derivative is related to the pseudogroup for the complex Möbius group. In fact if is a biholomorphism defined on then is a quadratic differential on . If is a bihomolorphism defined on and and are quadratic differentials on ; moreover is a quadratic differential on , so that is also a quadratic differential on . The identity

:$S(f\circ g) = g_*S(f) + S(g)$

is thus the analogue of a 1-cocycle for the pseudogroup of biholomorphisms with coefficients in holomorphic quadratic differentials. Similarly $\varphi_0(f) = \log f^\prime$ and $\varphi_1(f) = f^/f^\prime$ are 1-cocycles for the same pseudogroup with values in holomorphic functions and holomorphic differentials. In general 1-cocycle can be defined for holomorphic differentials of any order so that

:$\varphi(f\circ g) = g_*\varphi(f) + \varphi(g).$

Applying the above identity to inclusion maps , it follows that ; and hence that if is the restriction of , so that , then . On the other hand, taking the local holomororphic flow defined by holomorphic vector fields—the exponential of the vector fields—the holomorphic pseudogroup of local biholomorphisms is generated by holomorphic vector fields. If the 1-cocycle 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 :

:\varphi( ,Y = X \varphi(Y) - Y \varphi(X).

Restricting to the Lie algebra of polynomial vector fields with basis , 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 , whereas here it is just for a subalgebra acting on holomorphic (or polynomial) differentials of order . Again, assuming that vanishes on rotations of , there are non-zero 1-cocycles, unique up to scalar multiples. only for differentials of degree 0, 1 and 2 given by the same derivative formula

:$\varphi_k\left(p(z) \right) = p^(z) \, (dz)^k,$

where is a polynomial.

The 1-cocycles define the three pseudogroups by : this gives the scaling group (); the affine group (); and the whole complex Möbius group (). So these 1-cocycles are the special ordinary differential equations defining the pseudogroup. More significantly they can be used to define corresponding affine or projective structures and connections on Riemann surfaces. If is a pseudogroup of smooth mappings on , a topological space is said to have a -structure if it has a collection of charts that are homeomorphisms from open sets in to open sets in such that, for every non-empty intersection, the natural map from to lies in . This defines the structure of a smooth -manifold if consists of local diffeomorphims and a Riemann surface if —so that —and consists of biholomorphisms. If is the affine pseudogroup, is said to have an affine structure; and if is the Möbius pseudogroup, is said to have a projective structure. Thus a genus one surface given as for some lattice has an affine structure; and a genus surface given as the quotient of the upper half plane or unit disk by a Fuchsian group has a projective structure.

Gunning in 1966 describes how this process can be reversed: for genus , the existence of a projective connection, defined using the Schwarzian derivative ₂ 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 ).

See also

* Riccati equation

Notes

References

*, Chapter 6, "Teichmüller Spaces"
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*, Chapter 10, "The Schwarzian".
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*, Section 12, "Mapping of polygons with circular arcs".
*, "On the theory of generalized Lamé functions".
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*{{citation, last1=Takhtajan, first1= Leon A., last2=Teo, first2=Lee-Peng, title=Weil-Petersson metric on the universal Teichmüller space, series= Mem. Amer. Math. Soc., volume= 183 , year=2006, number=861

Projective geometry
Modular forms
Ordinary differential equations
Complex analysis
Conformal mappings