CONFORMAL UNIFORMIZATION OF PLANAR PACKINGS BY DISK PACKINGS DIMITRIOS NTALAMPEKOS

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CONFORMAL UNIFORMIZATION OF PLANAR PACKINGS BY

DISK PACKINGS

DIMITRIOS NTALAMPEKOS

Abstract. A Sierpi´nski packing in the 2-sphere is a countable collection of

disjoint, non-separating continua with diameters shrinking to zero. We show

that any Sierpi´nski packing by continua whose diameters are square-summable

can be uniformized by a disk packing with a packing-conformal map, a notion

that generalizes conformality in open sets. Being special cases of Sierpi´nski

packings, Sierpi´nski carpets and some domains and can be uniformized by disk

packings as well. As a corollary of the main result, the conformal loop ensemble

(CLE) carpets can be uniformized conformally by disk packings, answering a

question of Rohde–Werness.

Contents

1. Introduction 1

2. Preliminaries 6

3. Transboundary modulus estimates 14

4. Uniformization of Sierpi´nski packings 18

5. Topology of planar maps 30

6. Continuous extension 37

7. Homeomorphic extension 44

References 47

1. Introduction

One of the most intriguing open problems in complex analysis is Koebe’s conjec-

ture [Koe08], predicting that every domain in the Riemann sphere is conformally

equivalent to a circle domain, i.e., a domain whose complementary components are

geometric disks or points. This conjecture was established for ﬁnitely connected

domains by Koebe himself [Koe20] and it took over 70 years until it was estab-

lished for countably connected domains by He–Schramm [HS93]. This result was

proved with a diﬀerent method by Schramm [Sch95] in a seminal work, where the

notion of transboundary modulus is introduced. More recently, Rajala [Raj21] gave

another proof of the result, providing a new perspective. Remarkably, in [Sch95],

Schramm establishes Koebe’s conjecture for all cofat domains, i.e., domains whose

complementary components satisfy a uniform geometric condition that we discuss

Date: October 4, 2022.

2020 Mathematics Subject Classiﬁcation. Primary 30C20, 30C65; Secondary 30C35, 30C62.

Key words and phrases. Sierpi´nski carpet, Sierpi´nski packing, packing-conformal, packing-

quasiconformal, conformal loop ensemble, uniformization, Koebe’s conjecture.

The author is partially supported by NSF Grant DMS-2000096.

arXiv:2210.00164v1 [math.CV] 1 Oct 2022

2 DIMITRIOS NTALAMPEKOS

below in Section 2.2, independently of connectivity. The general case of Koebe’s

conjecture seems to be far out of reach. Koebe’s conjecture and uniformization

problems for domains in metric surfaces other than the Riemann sphere has been

studied in [MW13, RR21, Reh22].

A topic very closely related to Koebe’s conjecture is the uniformization of Sier-

pi´nski carpets. A Sierpi´nski carpet is a continuum in the sphere that has empty

interior and is obtained by removing from the sphere countably many open Jordan

regions, called peripheral disks, with disjoint closures and diameters shrinking to

zero. A fundamental result of Whyburn [Why58] states that all Sierpi´nski carpets

are homeomorphic to each other. Bonk [Bon11] proved that if the peripheral disks

of a Sierpi´nski carpet are uniformly relatively separated uniform quasidisks, then

the carpet can be mapped with a quasisymmetric map to a round carpet, i.e., a

carpet whose peripheral disks are geometric disks. Later, in [Nta20b] the author

developed a potential theory on Sierpi´nski carpets of area zero and proved that if

the peripheral disks of such a carpet are uniformly fat and uniformly quasiround,

then the carpet can be mapped in a natural way to a square carpet, deﬁned in the

obvious manner, with a map that is carpet-conformal in the sense that it preserves

a type of modulus. We note that the geometric assumptions in [Nta20b] are weaker

than in [Bon11]. However, if one strengthens the assumptions to uniformly rel-

atively separated uniform quasidisks, then the carpet-conformal map of [Nta20b]

is upgraded to a quasisymmetry. Both mentioned works of Bonk and the author

depend crucially on the notion of transboundary modulus of Schramm [Sch95].

In this work we push the results of [Bon11,Nta20b] to their limit and we remove

entirely the geometric assumptions at the cost of weakening the topological proper-

ties and the regularity of the uniformizing conformal map. Instead, we only impose

the square-summability of the diameters of the peripheral disks. Before stating the

results we give the required deﬁnitions.

Let {pi}i∈Nbe a collection of pairwise disjoint and non-separating continua in

the Riemann sphere b

Csuch that diam(pi)→0 as i→ ∞. The collection {pi}i∈N

is called a Sierpi´nski packing and the set X=b

C\Si∈Npiis its residual set. When

it does not lead to a confusion, we make no distinction between the terms packing

and residual set. The continua pi,i∈N, are called the peripheral continua of X.

Note that if the peripheral continua of Xare closed Jordan regions, then Xis a

Sierpi´nski carpet, provided that it has empty interior. Thus, Sierpi´nski packings

can be regarded as a generalization of Sierpi´nski carpets.

The natural spaces that can be used to parametrize a Sierpi´nski packing are

round Sierpi´nski packings, i.e., packings whose peripheral continua are (possibly

degenerate) closed disks. We now state our main theorem.

Theorem 1.1. Let Y=b

C\Si∈Nqibe a Sierpi´nski packing whose peripheral con-

tinua are closed Jordan regions or points with diameters in `2(N). Then there exist

(A) a collection of disjoint closed disks {pi}i∈N, where piis degenerate if and

only if qiis degenerate, a round Sierpi´nski packing X=b

C\Si∈Npi,

(B) a continuous, surjective, and monotone map H:b

C→b

Cwith the property

that H−1(int(qi)) = int(pi)for each i∈N, and

C),

with the following properties.

CONFORMAL UNIFORMIZATION BY DISK PACKINGS 3

•(Transboundary upper gradient inequality) There exists a curve family Γ0

in b

Cwith Mod2Γ0= 0 such that for all curves γ: [a, b]→b

Coutside Γ0we

have

σ(H(γ(a)), H(γ(b))) ≤Zγ

ρHds +X

i:pi∩|γ|6=∅

diam(qi)

•(Conformality) For each Borel set E⊂b

Cwe have

ZH−1(E)

ρ2

HdΣ≤Σ(E∩Y).

Moreover, if Yis cofat, then Hmay be taken to be a homeomorphism of the sphere.

Here σdenotes the spherical distance and Σ is the spherical measure. The mono-

toniciy of Hmeans that the preimage of every point is a continuum and is equivalent

to the statement that His the uniform limit of homeomorphisms; see Section 2.3.

The map Hin the conclusion of the theorem is called a packing-conformal map.

Our deﬁnition of a packing-conformal map is motivated by the transboundary mod-

ulus of Schramm and by the so-called analytic deﬁnition of quasiconformality for

maps between metric spaces [Wil12]. Moreover, an analogous deﬁnition under the

terminology weakly quasiconformal map has been used recently by Romney and

the author [NR22b, NR22a] in the solution of the problem of quasiconformal uni-

formization of spheres of ﬁnite area.

If Uis an open subset of b

Ccontained in the packing Y, then the map Hof

Theorem 1.1 is a conformal map in H−1(U) in the usual sense. However, not

every conformal map between domains satisﬁes the transboundary upper gradient

inequality. Nevertheless, one can show that this is always the case for countably

connected domains.

Remark 1.2.We remark that although H−1(int(qi)) = int(pi) in Theorem 1.1 (B),

the continuum H−1(qi) might be larger than the disk piwhen the packing Yis

not cofat. It is precisely this phenomenon that prevents us from proving Koebe’s

conjecture with this method. However, a non-trivial consequence of the topological

and regularity conditions of Theorem 1.1 is that the map His degenerate on the

set H−1(qi)\pi, in the sense that it maps each continuum E⊂H−1(qi)\pito a

point; see Figure 1. We prove this fact in Proposition 6.2.

In fact, Theorem 1.1 is a consequence of a more general uniformization theorem

for Sierpi´nski packings Ywithout the topological assumption that the peripheral

continua are closed Jordan regions or points. To each Sierpi´nski packing Ywe can

associate a topological sphere E(Y) by collapsing all peripheral continua to points,

in view of Moore’s decomposition theorem [Moo25].

Theorem 1.3. Let Y=b

C\Si∈Nqibe a Sierpi´nski packing such that the diameters

of the peripheral continua lie in `2(N). Then there exists a round Sierpi´nski packing

Xand a packing-conformal map from E(X)onto E(Y).

As we see, the uniformizing packing-conformal map exists only at the level of

the topological spheres E(X),E(Y), and in general does not induce a map between

the packings X, Y in the sphere b

C. For the deﬁnition of packing-conformal maps

between the associated topological spheres see Section 2.6. The above theorem is

restated as Corollary 4.2. The statement is proved via an approximation argument.

4 DIMITRIOS NTALAMPEKOS

Figure 1. The set H−1(qi) might be larger than the disk pi. In

this ﬁgure it contains the shaded regions and bold “branches”,

which are subsets of X. However, the map His constant in each

of them.

We consider the ﬁnitely connected domains Yn=b

C\Sn

i=1 qiand we uniformize

them conformally by ﬁnitely connected circle domains Xnusing Koebe’s theorem.

Then our task is to show that the conformal maps from Xnto Ynconverge in a

uniform sense to the desired limiting map from E(X) onto E(Y).

This proof strategy is also followed by Schramm [Sch95] in showing that co-

fat domains can be uniformized by circle domains and in [Bon11] in uniformizing

Sierpi´nski carpets by round carpets. The recent developments in the ﬁeld of analy-

sis on metric spaces and our much more thorough understanding of quasiconformal

maps between metric spaces allow us to identify the topological and regularity prop-

erties of the limiting map in our more fractal setting, where no uniform geometry

is imposed, as in the works of Schramm and Bonk.

We note that in unpublished work, Rohde and Werness [RW15] show that the

complementary disks of the circle domain Xnconverge in the Hausdorﬀ sense after

passing to a subsequence to a collection of pairwise disjoint disks. However, they

were not able to identify the limit of the conformal maps from Xnto Yn.

Theorem 1.1 is proved by showing that the topological assumptions on Yallow

one to lift a packing-conformal map between E(X) and E(Y) as in Theorem 1.3 to

the map Hin the sphere b

Cthat has the desired properties. This lifting process is

achieved through Theorem 6.1, which provides a monotone map H. In the case that

Yis cofat, the homeomorphism Has in the last part of Theorem 1.1 is provided by

Theorem 7.1. Thus, Theorem 1.1 is a consequence of Corollary 4.2, Theorem 6.1,

and Theorem 7.1.

Another generalization is that we do not need to restrict to round Sierpi´nski

packings Xin order to parametrize a given packing Y. Instead of using geometric

disks as the peripheral continua of X, one can use homothetic images of any count-

able collection of uniformly fat and non-separating continua, such as squares. See

Corollary 4.3 for the precise statement.

As a corollary of the main theorem we give an answer to a question of Rohde–

Werness [RW15] regarding the uniformization of the conformal loop ensemble (CLE)

carpet. CLE was introduced by Sheﬃeld–Werner [SW12], as a random collection

of Jordan curves in the unit disk that combines conformal invariance and a natural

CONFORMAL UNIFORMIZATION BY DISK PACKINGS 5

Figure 2. A CLE carpet (simulation by D.B. Wilson [Wil]).

restriction property; see Figure 2. Each CLE gives rise to a Sierpi´nski carpet with

non-uniform geometry; hence the current carpet uniformization theory of [Bon11,

Nta20b] is not suﬃcient to treat them. However, Rohde–Werness [RW15] proved

in unpublished work that, with probability 1, the diameters of the peripheral disks

of a CLE carpet are square-summable. Therefore, we obtain the following corollary

of the main theorem.

Corollary 1.4. If Yis a CLE carpet, almost surely there exists a round Sierpi´nski

packing Xand a packing-conformal map that maps Xonto Y.

It would be interesting to obtain some stronger statements for the uniformization

of CLE carpets. We pose several questions for further study.

Question 1.5.Under what conditions is the uniformizing round Sierpi´nski pack-

ing Xand the packing-conformal map Hof Theorem 1.1 unique (up to M¨obius

transformations)?

If one could at least show the uniqueness of X, then this would imply that CLE

gives rise to another stochastic process that generates round packings.

Question 1.6.Under what conditions is Xa carpet whenever Yis a carpet?

Theorem 1.1 already shows that a suﬃcient condition is the cofatness of Y. What

about CLE carpets?

Question 1.7.Can one use the present techniques to prove Koebe’s conjecture for

domains whose complementary components have diameters in `2(N)?

Another natural question is whether one can obtain alternative proofs of results

of [Bon11,Nta20b] upon strengthening the geometric assumptions on the peripheral

continua.

Question 1.8.If the peripheral continua of a packing Yare uniformly relatively

separated uniform quasidisks, is the map Hof Theorem 1.1 quasisymmetric?

Question 1.9.If the peripheral continua of a packing Yare uniformly fat and

uniformly quasiround, is the map Hof Theorem 1.1 carpet-conformal in the sense

of [Nta20b]?

In the subsequent paper [Nta22], we use the notion of packing-conformal maps

and the results of the present paper in order to study the problem of conformal rigid-

ity of circle domains, a problem that is closely related to the uniqueness in Koebe’s

conjecture. A circle domain is conformally rigid if every conformal map from that

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CONFORMALUNIFORMIZATIONOFPLANARPACKINGSBYDISKPACKINGSDIMITRIOSNTALAMPEKOSAbstract.ASierpinskipackinginthe2-sphereisacountablecollectionofdisjoint,non-separatingcontinuawithdiametersshrinkingtozero.WeshowthatanySierpinskipackingbycontinuawhosediametersaresquare-summablecanbeuniformizedbyadiskpackin...

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