Loewner differential equation
In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (conformal mappings of the open disk onto the complex plane with a curve joining 0 to ∞ removed), Loewner's method was later developed in 1943 by the Russian mathematician Pavel Parfenevich Kufarev (1909–1968). Any family of domains in the complex plane that expands continuously in the sense of Carathéodory to the whole plane leads to a one parameter family of conformal mappings, called a Loewner chain, as well as a two parameter family of holomorphic univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions on the disk with positive real part. The Loewner semigroup generalizes the notion of a univalent semigroup.
The Loewner differential equation has led to inequalities for univalent functions that played an important role in the solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner himself used his techniques in 1923 for proving the conjecture for the third coefficient. The Schramm–Loewner equation, a stochastic generalization of the Loewner differential equation discovered by Oded Schramm in the late 1990s, has been extensively developed in probability theory and conformal field theory.
Subordinate univalent functions
f is said to be subordinate to g if and only if there is a univalent mapping φ of D into itself fixing 0 such that
for |z| < 1.
A necessary and sufficient condition for the existence of such a mapping φ is that
Necessity is immediate.
Conversely φ must be defined by
By definition φ is a univalent holomorphic self-mapping of D with φ(0) = 0.
Since such a map satisfies 0 < |φ'(0)| ≤ 1 and takes each disk Dr, |z| < r with 0 < r < 1, into itself, it follows that
For 0 ≤ t ≤ ∞ let U(t) be a family of open connected and simply connected subsets of C containing 0, such that
if s < t,
Thus if ,
in the sense of the Carathéodory kernel theorem.
If D denotes the unit disk in C, this theorem implies that the unique univalent maps ft(z)
Moreover, the function is positive, continuous, strictly increasing and continuous.
By a reparametrization it can be assumed that
The univalent mappings ft(z) are called a Loewner chain.
The Koebe distortion theorem shows that knowledge of the chain is equivalent to the properties of the open sets U(t).
If ft(z) is a Loewner chain, then
for s < t so that there is a unique univalent self mapping of the disk φs,t(z) fixing 0 such that
By uniqueness the mappings φs,t have the following semigroup property:
for s ≤ t ≤ r.
They constitute a Loewner semigroup.
The self-mappings depend continuously on s and t and satisfy
Loewner differential equation
The Loewner differential equation can be derived either for the Loewner semigroup or equivalently for the Loewner chain.
For the semigroup, let
for |z| < 1.
Then w(t)=φs,t(z) satisfies the ordinary differential equation
with initial condition w(s) = z.
To obtain the differential equation satisfied by the Loewner chain ft(z) note that
so that ft(z) satisfies the differential equation
with initial condition
The Picard–Lindelöf theorem for ordinary differential equations guarantees that these equations can be solved and that the solutions are holomorphic in z.
The Loewner chain can be recovered from the Loewner semigroup by passing to the limit:
Finally given any univalent self-mapping ψ(z) of D, fixing 0, it is possible to construct a Loewner semigroup φs,t(z) such that
Similarly given a univalent function g on D with g(0) =0, such that g(D) contains the closed unit disk, there is a Loewner chain ft(z) such that
Results of this type are immediate if ψ or g extend continuously to ∂D. They follow in general by replacing mappings f(z) by approzimations f(rz)/r and then using a standard compactness argument.
Holomorphic functions p(z) on D with positive real part and normalized so that p(0) = 1 are described by the Herglotz representation theorem:
where μ is a probability measure on the circle. Taking a point measure singles out functions
with |κ(t)| = 1, which were the first to be considered by Loewner (1923).
Inequalities for univalent functions on the unit disk can be proved by using the density for uniform convergence on compact subsets of slit mappings. These are conformal maps of the unit disk onto the complex plane with a Jordan arc connecting a finite point to ∞ omitted. Density follows by applying the Carathéodory kernel theorem. In fact any univalent function f(z) is approximated by functions
which take the unit circle onto an analytic curve. A point on that curve can be connected to infinity by a Jordan arc. The regions obtained by omitting a small segment of the analytic curve to one side of the chosen point converge to g(D) so the corresponding univalent maps of D onto these regions converge to g uniformly on compact sets.
To apply the Loewner differential equation to a slit function f, the omitted Jordan arc c(t) from a finite point to ∞ can be parametrized by [0,∞) so that the map univalent map ft of D onto C less c([t,∞)) has the form
with bn continuous. In particular
For s ≤ t, let
with an continuous.
This gives a Loewner chain and Loewner semigroup with
where κ is a continuous map from [0,∞) to the unit circle.
To determine κ, note that φs,t maps the unit disk into the unit disk with a Jordan arc from an interior point to the boundary removed. The point where it touches the boundary is independent of s and defines a continuous function λ(t) from [0,∞) to the unit circle. κ(t) is the complex conjugate (or inverse) of λ(t):
Equivalently, by Carathéodory's theorem ft admits a continuous extension to the closed unit disk and λ(t), sometimes called the driving function, is specified by
Not every continuous function κ comes from a slit mapping, but Kufarev showed this was true when κ has a continuous derivative.
Application to Bieberbach conjecture
for the third coefficient of a univalent function
In this case, rotating if necessary, it can be assumed that a3 is non-negative.
with an continuous. They satisfy
the Loewner differential equation implies
which immediately implies Bieberbach's inequality
Since a3 is non-negative and |κ(t)| = 1,
using the Cauchy–Schwarz inequality.
- Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, ISBN 0-387-90795-5
- Kufarev, P. P. (1943), "On one-parameter families of analytic functions", Mat. Sbornik, 13: 87–118
- Lawler, G. F. (2005), Conformally invariant processes in the plane, Mathematical Surveys and Monographs, 114, American Mathematical Society, ISBN 0-8218-3677-3
- Loewner, C. (1923), "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I", Math. Ann., 89: 103–121, doi:10.1007/BF01448091
- Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, 15, Vandenhoeck & Ruprecht