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 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.
be holomorphic univalent functions on the unit disk
must be defined by By definition φ is a univalent holomorphic self-mapping of
, this theorem implies that the unique univalent maps
given by the Riemann mapping theorem are uniformly continuous on compact subsets of
The Koebe distortion theorem shows that knowledge of the chain is equivalent to the properties of the open sets
satisfies the ordinary differential equation with initial condition
To obtain the differential equation satisfied by the Loewner chain
The Loewner chain can be recovered from the Loewner semigroup by passing to the limit: Finally given any univalent self-mapping
contains the closed unit disk, there is a Loewner chain
Taking a point measure singles out functions with
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.
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
[2] To apply the Loewner differential equation to a slit function
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
admits a continuous extension to the closed unit disk and
comes from a slit mapping, but Kufarev showed this was true when
Loewner (1923) used his differential equation for slit mappings to prove the Bieberbach conjecture for the third coefficient of a univalent function In this case, rotating if necessary, it can be assumed that
They satisfy If the Loewner differential equation implies and So which immediately implies Bieberbach's inequality Similarly Since