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 holomorphicunivalent 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.
To obtain the differential equation satisfied by the Loewner chain note that
so that 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 .
The Loewner chain can be recovered from the Loewner semigroup by passing to the limit:
Finally given any univalent self-mapping of , fixing , it is possible to construct a Loewner semigroup such that
Similarly given a univalent function on with , such that contains the closed unit disk, there is a Loewner chain such that
Results of this type are immediate if or extend continuously to . They follow in general by replacing mappings by approximations and then using a standard compactness argument.[1]
Holomorphic functions on with positive real part and normalized so that are described by the Herglotz representation theorem:
where is a probability measure on the circle. Taking a point measure singles out functions
with , 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 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 so the corresponding univalent maps of onto these regions converge to uniformly on compact sets.[2]
To apply the Loewner differential equation to a slit function , the omitted Jordan arc from a finite point to can be parametrized by so that the map univalent map of onto less has the form
with continuous. In particular
For , let
with continuous.
This gives a Loewner chain and Loewner semigroup with
where is a continuous map from to the unit circle.[3]
To determine , note that 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 and defines a continuous function from to the unit circle. is the complex conjugate (or inverse) of :
Equivalently, by Carathéodory's theorem admits a continuous extension to the closed unit disk and , sometimes called the driving function, is specified by
Not every continuous function comes from a slit mapping, but Kufarev showed this was true when has a continuous derivative.
Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, ISBN0-387-90795-5
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, vol. 114, American Mathematical Society, ISBN0-8218-3677-3
Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht