=C\
n i=1
Di,
where the setsDi are pairwise disjoint closed Jordan regions. Suppose that
the boundaries ∂Di are k-quasicircles and the regions Di are s-relatively
separated fori =1, . . . , n. Then is a φ-Loewner region with φ =φn,s,k
only depending onn,s, andk.
In the proof we need a simple fact about the existence of subcontinua. Namely, if x ∈C, r >0, and E⊆C is a continuum with x ∈E and E\ B(x, r)= ∅, then there exists a subcontinuum E ⊆E with x ∈E, E ⊆ B(x, r), andE∩∂B(x, r)= ∅. Note that thenr≤diam(E)≤2r. So every continuum can be “cut to size” near each of its points.
To see that this statement is true let E be the connected component of
E∩B(x, r) containing x. ThenE is a closed subset of E with x∈E ⊆ B(x, r). If we hadE∩∂B(x, r)= ∅, thenE would be relatively open inE
and soE=E⊆B(x, r). This is impossible sinceE\B(x, r)= ∅. SoE is a continuum with the desired properties.
Proof of Proposition7.5 The proof is by induction on nwiths andkfixed. The induction beginningn=1 is covered by Proposition7.3(the requirement ofs-relative separation is vacuous in this case). For the induction step suppose thatn≥2 and that the statement is true for regions with the stated properties andn−1 complementary components.
We may assume thatDnis the complementary component ofwith small- est diameter d :=diam(Dn). Let E and F be arbitrary continua in with relative separation(E, F )≤t where t >0. We have to show that if ρ is an arbitrary non-negative Borel function on Cwith sufficiently small mass
ρ2d < m, wherem=m(n, s, k, t ) >0, then there exists a rectifiable path
γ inconnectingEandF withγ ρ ds <1.
By induction hypothesis we can findm1=m1(n, s, k, t ) >0 such that if
ρ2d < m1,
then there exists a rectifiable pathα in:=∪Dnthat connectsE andF and satisfies
α
A suitable constantmwill be found in the course of the proof. We make the preliminary choicem=m1. Then there exists a rectifiable path α in
connecting E andF satisfying (33). If α stays inside (with the possible exception of its endpoints), we can take γ =α. So we may assume that α
hitsDn. LetU be the Loewner collar aroundDn found in Lemma7.4. The idea now is to removeα∩Dnfromαand to connect suitable pieces ofα\Dn by a rectifiable pathβ in U such that βρ ds <1/2. A concatenation ofβ
with pieces ofα will then give a rectifiable pathγ inwithγ ρ ds <1 as desired.
For carrying out the details of this argument, we consider several cases. Letc=c(s, k) >0 be the constant from Lemma7.4withNsd(Dn)\Dn⊆U.
Case 1. NeitherEnorF is contained inN1
3cd(Dn).
We choose a closed, possibly degenerate, subpath α of α by starting at the endpointx ofαinEand traveling alongα until we first hitN1
6cd(Dn)at the pointx∈N1
6cd(Dn), say. SinceαmeetsDn, there exists such a pointx .
Thenα\ {x} ⊆\Dn=.
The setα∪E is a continuum that contains the point x, but that is not contained in N1
3cd(Dn) by our assumption in this case. So if we choose
r = 16cd, then (α ∪E)\B(x, r)= ∅. By the statement about the exis- tence of subcontinua discussed before the proof, we can find a continuum
E⊆α∪Ethat is contained inB(x, r)such that diam(E)≥r=16cd. Then
E⊆Ncd(Dn)∩⊆U.
In the same way, we choose a closed subpathαofαwith endpointsy∈F
andy∈N1
6cd(Dn)such thatα
\{y} ⊆. Again we can find a subcontinuum
Fofα∪F that is contained inNcd(Dn)∩⊆Usuch that diam(F)≥16cd. ThenE, F⊆Uand
dist(E, F)≤(2c+1)d≤(12+6/c)(diam(E)∧diam(F)).
The last inequality implies that(E, F)≤C(s, k). SinceU isφ-Loewner withφ =φs,k there exists a constant m2=m2(s, k) >0 with the following
property: if we impose the additional condition
ρ2d < m2
on ρ (as we may), then there exists a rectifiable path β in U ⊆ with
βρ ds <1/2 that connectsEandF. The pathβwill lie inwith the possi- ble exception of it endpoints. One endpoint ofβlies inE⊆α∪E⊆∪E, and one inF⊆α∪F ⊆∪F. So by concatenatingβ with suitable pieces
ofα andα, we obtain a rectifiable pathγ inwith γρ ds <1 that con- nectsEandF.
Case 2.t (diam(E)∧diam(F ))≥ 13cd.
We choose subpathsα andαofα as in Case 1. Arguing similarly as in this case, we can find continuaE⊆α∪EandF⊆α∪F withE, F⊆ Ncd(Dn)∩⊆U such that diam(E)≥ 13(diam(E)∧cd) and diam(F)≥
1
3(diam(F )∧cd). Then again we have
dist(E, F)≤(2c+1)d,
and also
diam(E)∧diam(F)≥1
3(diam(E)∧diam(F )∧cd)≥
cd
9(t∨1).
Hence
(E, F)≤C(s, k, t ).
In other words, the relative distance ofE andFis controlled bys,k, andt. By the Loewner property ofU we know that if
ρ2d < m3,
wherem3=m3(s, k, t ) >0, then there exists a rectifiable pathβ inU⊆
withβρ ds <1/2 that connectsEandF. As in Case 1, this leads to a path
γ as desired.
Case 3.EorF lies inN1
3cd(Dn), and we have
t (diam(E)∧diam(F )) <1
3cd. We may assumeE⊆N1
3cd(Dn). ThenE⊆Ncd(Dn)∩⊆U, and dist(E, F )≤t (diam(E)∧diam(F ))≤1
3cd.
Pick points x ∈ E and y ∈ F with σ (x, y) = dist(E, F ), and let r = 1
3(diam(F )∧cd). There exists a continuum F⊆F ∩B(y, r) with y∈F
and diam(F)≥r = 13(diam(F )∧cd). Then F ⊆Ncd(Dn)∩⊆U and dist(E, F)=σ (x, y)=dist(E, F ).
This implies that
dist(E, F)=dist(E, F )≤t (diam(E)∧diam(F )) <1
and so
dist(E, F)≤tdiam(E)∧tdiam(F )∧1
3cd≤3(t∨1)(diam(E)∧diam(F
)).
We conclude that(E, F)≤3(t∨1). SinceU is Loewner we know that if
ρ2d < m4,
wherem4=m4(s, k, t ) >0, then there exists a rectifiable pathβ inU⊆
withβρ ds <1 that connectsEandF⊆F. In this case we can takeγ =β. In conclusion, if
ρ2d < m,
wherem=min{m1, m2, m3, m4}, then we can find a rectifiable pathγ in
that connectsE andF and satisfiesγρ ds <1. Sincem >0 only depends onn, s, k, t, the statement follows.
8 Bounds for transboundary modulus
The present chapter is the technical core of the paper. We will prove various bounds for transboundary modulus. We use the chordal metricσ onCand the spherical measure. We will make repeated use of the relation(B(x, r))= (B(x, r))∼r2forx∈Cand small enoughr >0. Actually, we have
(B(x, r))=(B(x, r))=π r2 (34) valid for allx∈Cand 0< r≤2=diam(C).
A setM ⊆C is called λ-quasi-round, whereλ≥1, if there existx0∈C
and r ∈(0,diam(C)] =(0,2] such that B(x0, r/λ)⊆M ⊆B(x0, r). Note
that in this case diam(M)≥ r/λ. By Proposition 4.3 every Jordan region whose boundary is a quasicircle is quasi-round, quantitatively.