[Turkmath:2998] Fwd: [New post] 246C notes 3: Univalent functions, the Loewner equation, and the Bieberbach conjecture

[yilmaz akyildiz]

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Date: Thu, May 3, 2018 at 1:18 AM
Subject: [New post] 246C notes 3: Univalent functions, the Loewner
equation, and the Bieberbach conjecture
To: yilmaz.akyildiz at gmail.com

https://terrytao.wordpress.com/author/teorth/



Terence Tao posted: " We now approach conformal maps from yet another
perspective. Given an open subset $latex {U}&fg=000000$ of the complex
numbers $latex {{\bf C}}&fg=000000$, define a univalent function on $latex
{U}&fg=000000$ to be a holomorphic function $l"
Respond to this post by replying above this line
New post on *What's new*
<https://terrytao.wordpress.com/author/teorth/> 246C notes 3: Univalent
functions, the Loewner equation, and the Bieberbach conjecture
<https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/>
by
Terence Tao <https://terrytao.wordpress.com/author/teorth/>

We now approach conformal maps from yet another perspective. Given an open
subset [image: {U}] of the complex numbers [image: {{\bf C}}], define
a univalent
function <https://en.wikipedia.org/wiki/Univalent_function> on [image: {U}]
to be a holomorphic function [image: {f: U \rightarrow {\bf C}}] that is
also injective. We will primarily be studying this concept in the case
when [image:
{U}] is the unit disk [image: {D(0,1) := \{ z \in {\bf C}: |z| < 1 \}}].

Clearly, a univalent function [image: {f: D(0,1) \rightarrow {\bf C}}] on
the unit disk is a conformal map from [image: {D(0,1)}] to the image [image:
{f(D(0,1))}]; in particular, [image: {f(D(0,1))}] is simply connected, and
not all of [image: {{\bf C}}] (since otherwise the inverse map [image:
{f^{-1}: {\bf C} \rightarrow D(0,1)}] would violate Liouville's theorem).
In the converse direction, the Riemann mapping theorem tells us that every
simply connected proper subset [image: {V \subsetneq {\bf C}}] of the
complex numbers is the image of a univalent function on [image: {D(0,1)}].
Furthermore, if [image: {V}] contains the origin, then the univalent
function [image: {f: D(0,1) \rightarrow {\bf C}}] with this image becomes
unique once we normalise [image: {f(0) = 0}] and [image: {f'(0) > 0}]. Thus
the Riemann mapping theorem provides a one-to-one correspondence between
simply connected proper subsets of the complex plane containing the origin,
and univalent functions [image: {f: D(0,1) \rightarrow {\bf C}}] with [image:
{f(0)=0}] and [image: {f'(0)>0}]. We will focus particular attention on the
univalent functions [image: {f: D(0,1) \rightarrow {\bf C}}] with the
normalisation [image: {f(0)=0}] and [image: {f'(0)=1}]; such functions will
be called schlicht functions
<https://en.wikipedia.org/wiki/De_Branges%27s_theorem#Schlicht_functions>.

One basic example of a univalent function on [image: {D(0,1)}] is the Cayley
transform <https://en.wikipedia.org/wiki/Cayley_transform> [image: {z
\mapsto \frac{1+z}{1-z}}], which is a Möbius transformation from [image:
{D(0,1)}] to the right half-plane [image: {\{ \mathrm{Re}(z) > 0 \}}]. (The
slight variant [image: {z \mapsto \frac{1-z}{1+z}}] is also referred to as
the Cayley transform, as is the closely related map [image: {z \mapsto
\frac{z-i}{z+i}}], which maps [image: {D(0,1)}] to the upper half-plane.)
One can square this map to obtain a further univalent function [image: {z
\mapsto \left( \frac{1+z}{1-z} \right)^2}], which now maps [image: {D(0,1)}]
to the complex numbers with the negative real axis [image: {(-\infty,0]}]
removed. One can normalise this function to be schlicht to obtain the Koebe
function <https://www.encyclopediaofmath.org/index.php/Koebe_function>

[image: \displaystyle f(z) := \frac{1}{4}\left( \left( \frac{1+z}{1-z}
\right)^2 - 1\right) = \frac{z}{(1-z)^2}, \ \ \ \ \ (1)]

which now maps [image: {D(0,1)}] to the complex numbers with the
half-line [image:
{[1/4,\infty)}] removed. A little more generally, for any [image: {\theta
\in {\bf R}}] we have the *rotated Koebe function*

[image: \displaystyle f(z) := \frac{z}{(1 - e^{i\theta} z)^2} \ \ \ \ \ (2)]

that is a schlicht function that maps [image: {D(0,1)}] to the complex
numbers with the half-line [image: {\{ re^{-i\theta}: r \geq 1/4\}}]
removed.

Every schlicht function [image: {f: D(0,1) \rightarrow {\bf C}}] has a
convergent Taylor expansion

[image: \displaystyle f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots]

for some complex coefficients [image: {a_1,a_2,\dots}] with [image: {a_1=1}].
For instance, the Koebe function has the expansion

[image: \displaystyle f(z) = z + 2 z^2 + 3 z^3 + \dots = \sum_{n=1}^\infty
n z^n]

and similarly the rotated Koebe function has the expansion

[image: \displaystyle f(z) = z + 2 e^{i\theta} z^2 + 3 e^{2i\theta} z^3 +
\dots = \sum_{n=1}^\infty n e^{(n-1)\theta} z^n.]

Intuitively, the Koebe function and its rotations should be the ``largest"
schlicht functions available. This is formalised by the famous Bieberbach
conjecture <https://en.wikipedia.org/wiki/De_Branges%27s_theorem>, which
asserts that for any schlicht function, the coefficients [image: {a_n}]
should obey the bound [image: {|a_n| \leq n}] for all [image: {n}]. After a
large number of partial results, this conjecture was eventually solved by
de Branges <https://mathscinet.ams.org/mathscinet-getitem?mr=772434>; see
for instance this survey of Korevaar
<https://mathscinet.ams.org/mathscinet-getitem?mr=856290> or this survey of
Koepf <https://mathscinet.ams.org/mathscinet-getitem?mr=2281158> for a
history.

It turns out that to resolve these sorts of questions, it is convenient to
restrict attention to schlicht functions [image: {g: D(0,1) \rightarrow
{\bf C}}] that are *odd*, thus [image: {g(-z)=-g(z)}] for all [image: {z}],
and the Taylor expansion now reads

[image: \displaystyle g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots]

for some complex coefficients [image: {b_1,b_3,\dots}] with [image: {b_1=1}].
One can transform a general schlicht function [image: {f: D(0,1)
\rightarrow {\bf C}}] to an odd schlicht function [image: {g: D(0,1)
\rightarrow {\bf C}}] by observing that the function [image: {f(z^2)/z^2:
D(0,1) \rightarrow {\bf C}}], after removing the singularity at zero, is a
non-zero function that equals [image: {1}] at the origin, and thus (as [image:
{D(0,1)}] is simply connected) has a unique holomorphic square root [image:
{(f(z^2)/z^2)^{1/2}}] that also equals [image: {1}] at the origin. If one
then sets

[image: \displaystyle g(z) := z (f(z^2)/z^2)^{1/2} \ \ \ \ \ (3)]

it is not difficult to verify that [image: {g}] is an odd schlicht function
which additionally obeys the equation

[image: \displaystyle f(z^2) = g(z)^2. \ \ \ \ \ (4)]

Conversely, given an odd schlicht function [image: {g}], the formula (4)
<https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#g-eq>
uniquely determines a schlicht function [image: {f}].

For instance, if [image: {f}] is the Koebe function (1)
<https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#koebe>,
[image: {g}] becomes

[image: \displaystyle g(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \dots, \ \ \
\ \ (5)]

which maps [image: {D(0,1)}] to the complex numbers with two slits [image:
{\{ \pm iy: y > 1/2 \}}] removed, and if [image: {f}] is the rotated Koebe
function (2)
<https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#rotated>,
[image: {g}] becomes

[image: \displaystyle g(z) = \frac{z}{1- e^{i\theta} z^2} = z + e^{i\theta}
z^3 + e^{2i\theta} z^5 + \dots. \ \ \ \ \ (6)]

De Branges established the Bieberbach conjecture by first proving an
analogous conjecture for odd schlicht functions known as Robertson's
conjecture <https://mathscinet.ams.org/mathscinet-getitem?mr=1503286>. More
precisely, we have

*Theorem 1 (de Branges' theorem)* Let [image: {n \geq 1}] be a natural
number.

   - (i) (Robertson conjecture) If [image: {g(z) = b_1 z + b_3 z^3 + b_5
   z^5 + \dots}] is an odd schlicht function, then

   [image: \displaystyle \sum_{k=1}^n |b_{2k-1}|^2 \leq n.]
   - (ii) (Bieberbach conjecture) If [image: {f(z) = a_1 z + a_2 z^2 + a_3
   z^3 + \dots}] is a schlicht function, then

   [image: \displaystyle |a_n| \leq n.]

It is easy to see that the Robertson conjecture for a given value of [image:
{n}] implies the Bieberbach conjecture for the same value of [image: {n}].
Indeed, if [image: {f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots}] is schlicht,
and [image: {g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots}] is the odd schlicht
function given by (3)
<https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#gsqr>,
then from extracting the [image: {z^{2n}}] coefficient of (4)
<https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#g-eq>
we obtain a formula

[image: \displaystyle a_n = \sum_{j=1}^n b_{2j-1} b_{2(n+1-j)-1}]

for the coefficients of [image: {f}] in terms of the coefficients of [image:
{g}]. Applying the Cauchy-Schwarz inequality, we derive the Bieberbach
conjecture for this value of [image: {n}] from the Robertson conjecture for
the same value of [image: {n}]. We remark that Littlewood and Paley had
conjectured <https://mathscinet.ams.org/mathscinet-getitem?mr=1574687> a
stronger form [image: {|b_{2k-1}| \leq 1}] of Robertson's conjecture, but
this was disproved for [image: {k=3}] by Fekete and Szegö
<https://mathscinet.ams.org/mathscinet-getitem?mr=1574865>.

To prove the Robertson and Bieberbach conjectures, one first takes a
logarithm and deduces both conjectures from a similar conjecture about the
Taylor coefficients of [image: {\log \frac{f(z)}{z}}], known as the *Milin
conjecture*. Next, one continuously enlarges the image [image: {f(D(0,1))}]
of the schlicht function to cover all of [image: {{\bf C}}]; done properly,
this places the schlicht function [image: {f}] as the initial function [image:
{f = f_0}] in a sequence [image: {(f_t)_{t \geq 0}}] of univalent maps [image:
{f_t: D(0,1) \rightarrow {\bf C}}] known as a Loewner chain
<https://en.wikipedia.org/wiki/Loewner_differential_equation#Loewner_chain>.
The functions [image: {f_t}] obey a useful differential equation known as
the Loewner equation
<https://en.wikipedia.org/wiki/Loewner_differential_equation>, that
involves an unspecified forcing term [image: {\mu_t}] (or [image:
{\theta(t)}], in the case that the image is a slit domain) coming from the
boundary; this in turn gives useful differential equations for the Taylor
coefficients of [image: {f(z)}], [image: {g(z)}], or [image: {\log
\frac{f(z)}{z}}]. After some elementary calculus manipulations to
``integrate" this equations, the Bieberbach, Robertson, and Milin
conjectures are then reduced to establishing the non-negativity of a
certain explicit hypergeometric function, which is non-trivial to prove
(and will not be done here, except for small values of [image: {n}]) but
for which several proofs exist in the literature.

The theory of Loewner chains subsequently became fundamental to a more
recent topic in complex analysis, that of the Schramm-Loewner equation
<https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution> (SLE),
which is the focus of the next and final set of notes.

Read more of this post
<https://terrytao.wordpress.com/2018/05/02/246c-notes-3-univalent-functions-the-loewner-equation-and-the-bieberbach-conjecture/#more-10580>
*Terence Tao <https://terrytao.wordpress.com/author/teorth/>* | 2 May, 2018
at 2:18 pm | Tags: Bieberbach conjecture
<https://terrytao.wordpress.com/tag/bieberbach-conjecture/>, Loewner
equation <https://terrytao.wordpress.com/tag/loewner-equation/>, Univalent
functions <https://terrytao.wordpress.com/tag/univalent-functions/> |
Categories: 246C - complex analysis
<https://terrytao.wordpress.com/category/teaching/246c-complex-analysis/>,
math.AP <https://terrytao.wordpress.com/category/mathematics/mathap/>,
math.CA <https://terrytao.wordpress.com/category/mathematics/mathca/>,
math.CV <https://terrytao.wordpress.com/category/mathematics/mathcv/>,
Uncategorized <https://terrytao.wordpress.com/category/uncategorized/> |
URL: https://wp.me/p3qzP-2KE

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