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
Let
and
be holomorphic univalent functions on the unit disk
,
, with
.
is said to be subordinate to
if and only if there is a univalent mapping
of
into itself fixing
such that
![{\displaystyle \displaystyle {f(z)=g(\varphi (z))}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c451b59a2adc37d755ad1185d2dc7579d4fce5d)
for
.
A necessary and sufficient condition for the existence of such a mapping
is that
![{\displaystyle f(D)\subseteq g(D).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8978754d65db9efa723fd138f48b1eb41a9919a)
Necessity is immediate.
Conversely
must be defined by
![{\displaystyle \displaystyle {\varphi (z)=g^{-1}(f(z)).}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abf66e96256a27f7d4a9936b145ba8607a7a0b1d)
By definition φ is a univalent holomorphic self-mapping of
with
.
Since such a map satisfies
and takes each disk
,
with
, into itself, it follows that
![{\displaystyle \displaystyle {|f^{\prime }(0)|\leq |g^{\prime }(0)|}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6efb77e943dc3c0a5360952b0c39d49e1e67224d)
and
![{\displaystyle \displaystyle {f(D_{r})\subseteq g(D_{r}).}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11e54abcce553af67c8f9eaabfef4594fa4e20dd)
Loewner chain
For
let
be a family of open connected and simply connected subsets of
containing
, such that
![{\displaystyle U(s)\subsetneq U(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7531b2180bd36f792177cfa2fc3c98a7b4e760b4)
if
,
![{\displaystyle U(t)=\bigcup _{s<t}U(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f94f901ba9be89f9357d5edf6cb4d8345a82a2d2)
and
![{\displaystyle U(\infty )=\mathbb {C} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7aab548f0127736ad7c17ec8ec9c8722f7bc2b9)
Thus if
,
![{\displaystyle U(s_{n})\rightarrow U(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac89e9f9389392cdfed106696c390d507ecec29b)
in the sense of the Carathéodory kernel theorem.
If
denotes the unit disk in
, this theorem implies that the unique univalent maps
![{\displaystyle f_{t}(D)=U(t),\,\,\,f_{t}(0)=0,\,\,\,\partial _{z}f_{t}(0)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7b6efc17ab826fa57e551fe737b5bb38be18a0d)
given by the Riemann mapping theorem are uniformly continuous on compact subsets of
.
Moreover, the function
is positive, continuous, strictly increasing and continuous.
By a reparametrization it can be assumed that
![{\displaystyle f_{t}^{\prime }(0)=e^{t}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efc98a13866328c7c6578125dc9539544f4e57f3)
Hence
![{\displaystyle f_{t}(z)=e^{t}z+a_{2}(t)z^{2}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/17c2e69e6bbdf4b6f2c18d0e5da0155a4b74bb9c)
The univalent mappings
are called a Loewner chain.
The Koebe distortion theorem shows that knowledge of the chain is equivalent to the properties of the open sets
.
Loewner semigroup
If
is a Loewner chain, then
![{\displaystyle \displaystyle {f_{s}(D)\subsetneq f_{t}(D)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68dc6858602216cf37fe7c9b198e000fe83efee3)
for
so that there is a unique univalent self mapping of the disk
fixing
such that
![{\displaystyle \displaystyle {f_{s}(z)=f_{t}(\varphi _{s,t}(z)).}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a110ed60a80fcf45189a7240acf9409f1ea766d8)
By uniqueness the mappings
have the following semigroup property:
![{\displaystyle \displaystyle {\varphi _{t,r}\circ \varphi _{s,t}=\varphi _{s,r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fb53b716ad7d23b9420e8d9eb77cd492b8d3d3)
for
.
They constitute a Loewner semigroup.
The self-mappings depend continuously on
and
and satisfy
![{\displaystyle \displaystyle {\varphi _{t,t}(z)=z.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e92f76b57dfd5d4b68b5484d125bc637813d23c)
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
![{\displaystyle \displaystyle {w_{s}(z)=\partial _{t}\varphi _{s,t}(z)|_{t=s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d29f8bf3ed2bcf6d5d77090be1c4f6d951c8ded2)
then
![{\displaystyle \displaystyle {w_{s}(z)=-zp_{s}(z)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18be1d381da40bde2a48330dee72f5c1fd7c2c4c)
with
![{\displaystyle \displaystyle {\Re \,p_{s}(z)>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e221c582c9188a333a98684cbc3e8e6ebce0b8)
for
.
Then
satisfies the ordinary differential equation
![{\displaystyle \displaystyle {{dw \over dt}=-wp_{t}(w)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b82dbe2b885a30597e39975e52a571a00c258531)
with initial condition
.
To obtain the differential equation satisfied by the Loewner chain
note that
![{\displaystyle \displaystyle {f_{t}(z)=f_{s}(\varphi _{s,t}(z))}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d75f346e647a45d753c05f0242f6bd25da14cd5)
so that
satisfies the differential equation
![{\displaystyle \displaystyle {\partial _{t}f_{t}(z)=zp_{t}(z)\partial _{z}f_{t}(z)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96323ef89e4edea99c3a689a6de0025c6f972b00)
with initial condition
![{\displaystyle \displaystyle {f_{t}(z)|_{t=0}=f_{0}(z).}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96d0c879df7063aad88ab0de01e16aa6aeeab80d)
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:
![{\displaystyle \displaystyle {f_{s}(z)=\lim _{t\rightarrow \infty }e^{t}\phi _{s,t}(z).}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e9155958c8b3c67210ff2b54912a828d777278)
Finally given any univalent self-mapping
of
, fixing
, it is possible to construct a Loewner semigroup
such that
![{\displaystyle \displaystyle {\varphi _{0,1}(z)=\psi (z).}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3ff06076a64abcdc61dc49761500983177117d)
Similarly given a univalent function
on
with
, such that
contains the closed unit disk, there is a Loewner chain
such that
![{\displaystyle \displaystyle {f_{0}(z)=z,\,\,\,f_{1}(z)=g(z).}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dca9c1885ebc8d565dcc789b08bc187bbdf49411)
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]
Slit mappings
Holomorphic functions
on
with positive real part and normalized so that
are described by the Herglotz representation theorem:
![{\displaystyle \displaystyle {p(z)=\int _{0}^{2\pi }{1+e^{-i\theta }z \over 1-e^{-i\theta }z}\,d\mu (\theta ),}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb121606276c311bde1a5637819fea77bcee372)
where
is a probability measure on the circle. Taking a point measure singles out functions
![{\displaystyle \displaystyle {p_{t}(z)={1+\kappa (t)z \over 1-\kappa (t)z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e86090287150e0c1323eaaad4927a404be35f32)
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
![{\displaystyle \displaystyle {g(z)=f(rz)/r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4783317bbee25a6bb643e3954c43af92cc586fed)
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
![{\displaystyle \displaystyle {f_{t}(z)=e^{t}(z+b_{2}(t)z^{2}+b_{3}(t)z^{3}+\cdots )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a22ac9b86fb8a2f91182c24950f4b3a788b2d6)
with
continuous. In particular
![{\displaystyle \displaystyle {f_{0}(z)=f(z).}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e50fac6aec117784089292532bcfba40dd862175)
For
, let
![{\displaystyle \displaystyle {\varphi _{s,t}(z)=f_{t}^{-1}\circ f_{s}(z)=e^{s-t}(z+a_{2}(s,t)z^{2}+a_{3}(s,t)z^{3}+\cdots )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e93f85393d9ae18dc49df79cae360e4ec34b628)
with
continuous.
This gives a Loewner chain and Loewner semigroup with
![{\displaystyle \displaystyle {p_{t}(z)={1+\kappa (t)z \over 1-\kappa (t)z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e86090287150e0c1323eaaad4927a404be35f32)
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
:
![{\displaystyle \displaystyle {\kappa (t)=\lambda (t)^{-1}.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01f0d0194cd4288598c883792335ef06964bb307)
Equivalently, by Carathéodory's theorem
admits a continuous extension to the closed unit disk and
, sometimes called the driving function, is specified by
![{\displaystyle \displaystyle {f_{t}(\lambda (t))=c(t).}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea719e9c5dc39938c82dfbbdc670d8c2a52a7e30)
Not every continuous function
comes from a slit mapping, but Kufarev showed this was true when
has a continuous derivative.
Application to Bieberbach conjecture
Loewner (1923) used his differential equation for slit mappings to prove the Bieberbach conjecture
![{\displaystyle \displaystyle {|a_{3}|\leq 3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e62093c29560a003b2e16e526d6b56f669e70544)
for the third coefficient of a univalent function
![{\displaystyle \displaystyle {f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b19a8f52da573fef2df1c0010f4dff90bc5f05)
In this case, rotating if necessary, it can be assumed that
is non-negative.
Then
![{\displaystyle \displaystyle {\varphi _{0,t}(z)=e^{-t}(z+a_{2}(t)z^{2}+a_{3}(t)z^{3}+\cdots )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3770ca8208aa05bbbeda05d52f3bccc20a565e4c)
with
continuous. They satisfy
![{\displaystyle \displaystyle {a_{n}(0)=0,\,\,a_{n}(\infty )=a_{n}.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86238dfa0319e6c913edd206a1a1187536aecec5)
If
![{\displaystyle \displaystyle {\alpha (t)=e^{-t}\kappa (t),}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea51c8316d97896c78cdad56f05d06dc56f631df)
the Loewner differential equation implies
![{\displaystyle \displaystyle {{\dot {a}}_{2}=-2\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c77a9d0e76e246377519d945eb60c91fee4eeae)
and
![{\displaystyle \displaystyle {{\dot {a}}_{3}=-2\alpha ^{2}-4\alpha \,a_{2}.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad89094c00b8f9e9fa34aeff01730daab173d92)
So
![{\displaystyle \displaystyle {a_{2}=-2\int _{0}^{\infty }\alpha (t)\,dt}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/513fd4fde1769957643de47815175030f2ee5fc9)
which immediately implies Bieberbach's inequality
![{\displaystyle \displaystyle {|a_{2}|\leq 2.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0e48e21a6dbebb26bbc482f2c5f43036c2c139)
Similarly
![{\displaystyle \displaystyle {a_{3}=-2\int _{0}^{\infty }\alpha (t)^{2}\,dt+4\left(\int _{0}^{\infty }\alpha (t)\,dt\right)^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c519951d9472c32012a7c4cfd3bdb2f3879551f)
Since
is non-negative and
,
![{\displaystyle \displaystyle {|a_{3}|=2\int _{0}^{\infty }|\Re \alpha (t)^{2}|\,dt+4\left(\int _{0}^{\infty }\Re \alpha (t)\,dt\right)^{2}}\leq 2\int _{0}^{\infty }|\Re \alpha (t)^{2}|\,dt+4\left(\int _{0}^{\infty }e^{-t}\,dt\right)\left(\int _{0}^{\infty }e^{t}(\Re \alpha (t))^{2}\,dt\right)=1+4\int _{0}^{\infty }(e^{-t}-e^{-2t})(\Re \kappa (t))^{2}\,dt\leq 3,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/741f710763d5efd9d50ea7a70d308ea54537269a)
using the Cauchy–Schwarz inequality.
Notes
References
- Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 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, vol. 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, hdl:10338.dmlcz/125927
- Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht