EXPLICIT MULTI-SLIT LOEWNER FLOWS AND THEIR GEOMETRY E. K. THEODOSIADIS

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EXPLICIT MULTI-SLIT LOEWNER FLOWS AND THEIR

GEOMETRY

E. K. THEODOSIADIS

Abstract. In this paper we present explicit solutions to the radial and chordal

Loewner PDE and we make an extensive study of their geometry. Speciﬁcally,

we study multi-slit Loewner ﬂows, driven by the time-dependent point masses

µt:“řn

j“1bjδtζjeiatuin the radial case and νt:“řn

j“1bjδtkj?1´tuin the

chordal case, where all the above parameters are chosen arbitrarily.

Furthermore, we investigate their close connection to the semigroup theory

of holomorphic functions, which also allows us to map the chordal case to the

radial one.

1. Introduction

Loewner’s theory was pioneered in 1923 by Charles Loewner (1893–1968), who

studied the properties of continuously evolving families of slit mappings of the unit

disc and discovered that such a family satisﬁes the so-called Loewner PDE. A few

years earlier, in 1916, Ludwig Bieberbach proved that the second coeﬃcient of the

Taylor series of a univalent (analytic and one-to-one) mapping of the unit disc,

satisﬁes the inequality |a2| ď 2|a1|, while he also conjectured that |an| ď n|a1|, for

every ně1. The ﬁrst theoretical application of Loewner’s PDE appeared in the

proof of the Bieberbach conjecture for the third coeﬃcient. An aﬃrmative answer

to the Bieberbach conjecture was ﬁrst given by Louis de Branges in 1984, however,

a second and simpler proof by Pommerenke and Fitzgerald in 1985 was possible

due to Loewner’s theory for slit mappings.

A contemporary and more general version of Loewner’s PDE was introduced by

Pavel Kufarev (1909–1968) in 1947. According to his study, given a continuous and

increasing family of simply connected domains, then their corresponding Riemann

maps, satisfy a PDE driven by an analytic function of the unit disc with positive real

part and conversely, any such PDE produces a family of solutions, whose images

form a continuous and increasing family of simply connected domains. Roughly

speaking, the driving functions are replaced by driving measures, since analytic

functions with positive real part are represented by ﬁnite Borel measures. This

generalizes Loewner’s theory to non-slit mappings. For this reason, nowadays, we

often refer to the Loewner-Kufarev PDE.

The radial Loewner PDE. Considering a simply connected domain Din

the Riemann sphere ˆ

Cwhere we omit at least two points, the Riemann mapping

theorem guarantees a conformal map from the unit disk onto D. Assuming a point

aPD, this map is unique when we require fp0q “ aand f1p0q ą 0.

Let pDtqtě0be a decreasing (resp. increasing), i.e, DtĂDsfor săt(resp.

DsĂDt), family of simply connected domains in ˆ

C, that is continuous in the sense

of Caratheodory’s kernel convergence. A decreasing (resp. increasing) Loewner

chain is deﬁned to be the family of the unique Riemann maps described above,

Date: December 27, 2022.

Key words and phrases. Loewner ﬂows, Riemann maps, Semigroups of holomorphic maps,

PDEs in the complex plane.

arXiv:2210.11396v2 [math.AP] 24 Dec 2022

2 ELEFTHERIOS THEODOSIADIS

ft:“fp¨, tq:DÑDt. We refer to fas the Loewner ﬂow. Assuming that the origin

is contained in each Dt, then ftexpands in Taylor series as

fpz, tq “ βptqz`. . . , |z| ă 1, t ě0

where βis decreasing (resp. increasing) with time, as follows by the Schwarz

lemma. For the rest, we shall only consider decreasing families. By monotonicity,

it also follows that βis almost everywhere diﬀerentiable. For any such family, there

exists a family of bounded Borel measures pµtqtě0on the unit circle, with mass

µtpBDq“´β1ptq{βptq, so that the Loewner-Kufarev equation

(1.1) Bf

Btpz, tq“´zf 1pz, tqżBD

ζ`z

ζ´zdµtpζq:“ ´zf 1pz, tqppz, tq

is satisﬁed for all zPDand for almost all tě0. In the classical notation, βptq

is given by the exponential function. This can be considered by reparameterizing

time, since βis monotonic. For a detailed approach, see [4] and [17].

Conversely, equation p1.1qadmits a unique solution for given initial values fpz, 0q.

We are interested in the initial value problem p1.1qand when fpz, 0q “ z, thus the

starting domain is the unit disk. Then, the solution fpz, tq:DÑDt“DzKt

describes a decreasing Loewner ﬂow. We refer to pKtqtě0as the growing hulls of

the evolution.

In terms of the transition functions φs,tpzq “ φpz;s, tq “ f´1

s˝ftpzqfor săt

the Loewner-Kufarev ODE is written as

(1.2) Bφ

Bspz;s, tq “ φpz;s, tqppφpz;s, tq, sq, φpz;t, tq “ z

for zPDand 0 ăsăt. Clearly, it is easier to solve the ODE instead of the

corresponding PDE and it will play the basic role in the forthcoming discussion.

The chordal Loewner PDE. The situation in the upper half plane is somewhat

similar to the radial case. Here, we consider a decreasing family of simply connected

domains pDtqtě0, such that DtĂH,Kt:“HzDtis compact and moreover HzKt

is also simply connected. Such a family of domains is produced by the so-called

chordal Loewner equation.

Let Dbe a simply connected domain as above. Then, by Riemann’s mapping

theorem there exists a conformal map gfrom Donto H, such that gpzq´zÑ0 as

zbecomes inﬁnite. This property is referred in the literature as the hydrodynamic

condition of g. The chordal Loewner ﬂows are always normalized such that each

map of the ﬂow satisﬁes the hydrodynamic condition.

The slit case in the upper half plane is treated as in the radial Loewner’s slit

case. A recent, detailed proof is given by A. Monaco and P. Gumenyuk in [15]. See

also A. Starnes [24] and M.- N. Technau [25] for the multiple and inﬁnitely many

slits versions respectively. Assume that γis a Jordan curve emanating from R,

with parameterization γ“γptq, 0 ďtă 8 and write Kt:“γpr0, tsq. Let gtbe the

corresponding Riemann maps described above and let ftbe the inverse mappings.

It is then proved that λptq:“gtpγptqq is a continuous real-valued function of tand

for each Tą0 we have the chordal Loewner ODE

Btpz, tq “ 2

gpz, tq ´ λptq, gpz, 0q “ 0

for all zPHzγpr0, T sq and 0 ďtďT. By taking its inverse, the chordal Loewner

PDE is written as Bf

Btpz, tq “ ´f1pz, tq2

z´λptq

in Hˆ r0,8q. The continuous function λis called the driving function of the ﬂow

fpz, tq.

EXPLICIT MULTI-SLIT LOEWNER FLOWS AND THEIR GEOMETRY 3

More generally, given a family of probability measures pµtqtě0of Rwith compact

support, we consider the ODE in Hˆ r0,8q

Btpz, tq “ żR

dµtpxq

wpz, tq ´ x

with initial value wpz, 0q “ 0. Let Tzbe the supremum of all t, such that the

solution to the equation is well deﬁned and gtpzq P Hfor all tďTz. Then, there

exists a unique solution gtwhich is conformal in the domain Dt:“ tz:Tzątu,

satisfying the hydrodynamic condition. Finally, the Loewner ﬂow ft“g´1

tsatisﬁes

the PDE Bf

Btpz, tq“´f1pz, tqżR

dµtpxq

z´x

for all zPHand tě0, called the chordal Loewner PDE. We refer to the compact

sets Kt:“HzDtas the compact hulls generated by the ﬂow. See [9] for details.

Connection to semigroups. There is a close connection of Loewner’s theory

to the semigroup theory for holomorphic self-maps of the unit disc. In fact, due to

the Berkson-Porta formula (see [2]), it turns out that a Loewner ﬂow fpz, tqdriven

by a time-independent function ppz, tq “ ppzq, forms a semigroup with ﬁxed point

the origin and it is parameterized as

fpz, tq “ h´1pe´thpzqq

for all zPDand tě0, where his a starlike function with respect to zero, called

the Koenigs function. In [23], A. Sola presents examples of Loewner ﬂows, driven

by time-independent densities.

To be more precise, a continuous elliptic semigroup of holomorphic self-maps of

the unit disc, with Denjoy-Wolﬀ point 0, is deﬁned as a family pφtqtě0ĂHpDqso

that

(1) φ0“idD

(2) φpz, t `sq “ φpφpz, tq, sq, for all zPDand s, t ě0,

(3) φtpzq Ñ 0, as tÑ 8, for all zPD,

(4) Dˆ r0,8q Q pz, tq ÞÑ φpz, tqis continuous in the uniform on compacts

topology.

For the rest, we shall only refer to continuous semigroups, thus (4) is always

assumed. It is proved that there exists some λPC, with Reλě0, so that

φ1p0q “ e´λt. We call λthe spectral value of the semigroup. Abbreviating the

deﬁnition, we might refer to e´λt as the spectral value instead. See proposition

8.1.4 in [2] for more details on the Denjoy-Wolﬀ theorem and the spectral value.

Now, given a semigroup pφtqtě0, then and only then, there exists a unique vector

ﬁeld GPHpDq, such that the PDE

Bφ

Btpz, tq “ Gpφpz, tqq

is satisﬁed in DˆT, where Tis an interval containing r0,8q. We call Gthe

inﬁnitesimal generator of the semigroup. The Berkson-Porta theorem (or formula)

states that given a non-constant GPHpDq, then Gis the inﬁnitesimal generator of

a semigroup if and only if, there exists some pPHpDqwith Repppzqq ą 0 in D, so

that

Gpzq“´zppzq.

In the Loewner language, pis the driving function as we previously mentioned.

A semigroup of the upper half plane is deﬁned similarly. Moreover, a non-elliptic

semigroup is deﬁned as above, but it has Denjoy-Wolﬀ point in the boundary. Thus,

there exists some τP BD, such that condition p3qis written as φtpzq Ñ τ.

4 ELEFTHERIOS THEODOSIADIS

To conclude, we see h, the Koenigs function of the semigroup, as the solution to

the ODE

Gpzq “ ´λhpzq

h1pzq

in D, where λis the spectral value of the semigroup. Observe by Berkson-Porta’s

formula and the characterization of spirallike functions (see next section), that

by taking the real part of the preceding equation, then his an Argpλq-spirallike

function of D. For our purposes, it is enough to view the Koenigs function as

discussed above. We may also think of it as the function that maps the orbits

tφpz, tq{tě0uonto logarithmic spirals (for elliptic semigroups), or onto half-lines

(for non-elliptic semigroups). Of course, the theory for Koenigs function is much

deeper. For a general discussion, see chapter 9 in [2].

Brief overview of literature. Loewner proved that growing curves correspond

to continuous driving functions, but the converse is not true. Kufarev presented an

example of point masses with the corresponding ﬂow being non-slit. For instance,

the driving function e´iλptq“kptq “ pe´t`i?1´e´2tq3produces the family of

domains Dt, that are Dminus the part of the disc lying in Dand intersecting

BDorthogonally, at the points kptqand kptq1

3. In the literature, we ﬁnd suﬃcient

conditions for a driving function to produce hulls that are curves. In particular, D.

Marshall and S. Rohde prove in [14], that Lip-1

2driving functions, with suﬃciently

bounded Lipp1

2q-norm, produce quasi-slit domains, while J. Lind, proved in her work

in [11], C0“4 to be the optimal upper bound for the norm. Although the preceding

results refer to single-slit ﬂows, a generalization for multiple-slit ﬂows was done by

S. Schleissinger in [22], proving that Lip-1

2driving functions λ1, . . . , λnproduce n

disjoint Jordan curves.

Some explicit ﬂows, that are relevant to our work as well, are found in [7] by

P. Kadanoﬀ, B. Nienhuis and W. Kager, where they explicitily solve the PDE for

the driving function k?1´tand they describe the geometry of the solutions for

the cases |k| ă 4 and |k| ą 4. Other cases of slit mappings in the upper half plane

are presented in [18] by D. Prokhorov, A. Zakharov and A. Zherdev. Since the

geometry of the slits will be the main topic of this text, we must note the work of

C. Wong [26], according to which, a driving function that is Lipschitz continuous

with exponent in p1

2,1sproduces a curve that grows vertically from the real line.

Our aim in this work is to present explicitily given, multi-slit Loewner ﬂows

both in the disc and in the upper half plane, by solving their corresponding PDE’s.

Therefore, the ﬁrst step in our study, is to write down the Riemann maps produced

by particularly chosen driving functions and the second step is to describe their

geometry. Finally, we present the close relation of the particular maps to semigroup

theory for holomorphic self maps of the unit disc, which allows to visualize a speciﬁc

class of Loewner PDE’s. We outline the structure of the text below.

2. Outline of the paper and preliminaries

We begin by presenting a couple of preliminary results about sprirallike domains,

that are necessary for the main ideas of the third and fourth section of this text. A

logarithmic spiral of angle ψP p´π

2,π

2qin the complex plane is deﬁned as the curve

with parameterization S:w“w0expp´e´iψtq,´8 ď tď 8, for some complex

number w0‰0.

Deﬁnition 1. A simply connected domain D, that contains the origin, is said to

be ψ-spirallike (with respect to zero), if for any point w0PD, the logarithmic spiral

S:w“w0expp´eiψtq,0ďtď 8 is contained in D.

EXPLICIT MULTI-SLIT LOEWNER FLOWS AND THEIR GEOMETRY 5

Deﬁnition 2. A univalent function fPHpDq, with fp0q “ 0, is said to be ψ-

spirallike if it maps the unit disc onto a ψ-spirallike domain D.

Note that 0-spirals are straight lines emanating from the origin and expanding to

inﬁnity. We refer to 0-spirallike domains/functions as starlike domains/functions.

The following theorem gives an analytic characterization of spirallike mappings. A

detailed proof is found in [4], paragraph 2.7.

Theorem 3. Let fPHpDq, with f1p0q ‰ 0and fpzq “ 0if and only if z“0.

Then fis ψ-spirallike, if and only if

Re ˆe´iψ zf 1pzq

fpzq˙ą0

for all zPD.

Although spirallike functions usually refer to analytic functions of the disc, we

can easily transfer the preceding deﬁnition on functions of the upper half plane.

Deﬁnition 4. A univalent function fPHpHq, with fpβq “ 0for some βPH, is

said to be ψ-spirallike (with respect to β), if it maps the upper half plane onto a

ψ-spirallike domain D.

Proposition 5. Let fPHpHq, with f1pβq ‰ 0and fpzq “ 0if and only if z“β.

Then, fis ψ-spirallike, if and only if

Im ˆe´iψ pz´βqpz´¯

βqf1pzq

fpzq˙ą0

for all zPH.

Proof. The result follows immediately from the disc case, by applying the M¨obius

transform T z “z´β

z´β, which maps Honto D.

Basic results. We start oﬀ by setting our conﬁguration in the unit disc. Given

some arbitrarily chosen points ζ1, . . . , ζnP BD, the weights b1, . . . , bną0 and the

exponent aPR, the role of the driving measures of the ﬂow will be played by the

measures

µt:“

k“1

bkδteiatζku.

Note that the ﬂow is not normalized, since we do not demand the measures to be

probablities; i.e, b1` ¨¨¨ ` bn“1. Our ﬁrst result in paragraph 3.3 is outlined

below.

Theorem 6. Assume the conﬁguration above and consider the radial Loewner-

Kufarev PDE in Dˆ r0,8q,

Btpz, tq “ ´f1pz, tqz

k“1

eiatζk`z

eiatζk´z

with initial value fpz, 0q “ z. Then, the Loewner ﬂow is of the form

fapz, tq “ φ´1pe´přn

k“1bk´iaqtφpe´iatzqq,

where φis an pArccotpa

řn

k“1bkq ´ π

2q-spirallike function of D.

Furthermore, for each k, with 1ďkďn, the trace ˆγpkq

a:“ tfapeiatζk, tq{ tě0u

is a smooth curve lying in Dthat starts perpendicularly from ζk, spiralling about

the origin.

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EXPLICITMULTI-SLITLOEWNERFLOWSANDTHEIRGEOMETRYE.K.THEODOSIADISAbstract.InthispaperwepresentexplicitsolutionstotheradialandchordalLoewnerPDEandwemakeanextensivestudyoftheirgeometry.Specically,westudymulti-slitLoewnerows,drivenbythetime-dependentpointmassest:°nj1bjtjeiatuintheradialcaseandt:°n...

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