∂D0, . . . , ∂Dnares-relatively separatedk-quasicircles, and that diam(∂D0)∧diam(∂Dn)≥d >0.
Then there exist constants C1=C1(s, k) >0, C2=C2(s, k, d) >0, and 0= 0(s, k, d)∈(0,2π )such that
C1≤hA≤C2, (56)
and
(Qi)≤2π− 0 for alli=1, . . . , n−1. (57)
Proof Let V =C \(D0 ∪ Dn), K = {Di : i = 1, . . . , n −1}, and =
(∂D0, ∂Dn;V ). We can extend the mapf in Corollary9.13(non-uniquely) to a homeomorphism fromV ontoA. By the properties of the mapf we then have
f ()=(∂iA, ∂oA;A).
Hence by invariance of transboundary modulus and Proposition11.2we get
MV ,K()=2π/ hA.
This shows that in order to establish inequality (56), it suffices to show that
MV ,K() is bounded below by a positive constant only depending on s, k andd, and bounded above by a constant only depending ons andk.
To produce the first bound note that by Lemma4.8the regionsD0, . . . , Dn are also s-relatively separated. So by Proposition 7.5 the region V =C\ (D0∪Dn)is φ-Loewner, where φ=φs,k. Moreover, for the continua ∂D0 and∂Dnwe have
(∂D0, ∂Dn)≤2/d.
Since the continua in K are s-relatively separated, and also λ-quasi- round withλ=λ(k)by Proposition4.3, it follows from Proposition8.1that
MV ,K()≥C(s, k, d) >0 as desired.
To produce an inequality in the opposite direction, note that
since∂D0and∂Dnares-relatively separated. Hence by Proposition8.4,
MV ,K()≤C(s, k). The first part of the theorem follows.
To prove the second part of the proposition consider one of theC∗-squares
Q1, . . . , Qn−1, say Q1. Under the mapf it corresponds to one of the Jor-
dan regionsD1, . . . , Dn−1, say toD1. LetV=C\(D0∪D1∪Dn). Then again by Proposition7.5the regionV isφ-Loewner withφ=φs,k. We can again invoke Proposition8.1and the invariance of transboundary modulus to conclude that
MU,L((∂iA, ∂oA;U ))=MV,K((∂D0, ∂Dn;V))≥C(s, k, d) >0. HereU=A\Q1,L= {Q2, . . . , Qn−1}, andK= {D2, . . . , Dn−1}.
On the other hand, suppose thatA= {z∈C:r <|z|< R}. Without loss of generality we may assume that
Q1= {seit:r≤s≤R, t ∈ [α,2π−α]},
wherer < r< R< Randα∈(0, π ). Then(Q1)=2(π−α)=log(R/r).
We have to show thatα cannot be smaller than a positive constant only de- pending ons,k, andd.
Note that every pathγ ∈=(∂iA, ∂Ao;U )lies in the complement of
Q1 and meets both circles{z∈C: |z| =r}and{z∈C: |z| =R}. Henceγ
passes through the channel
M= {seit :r< s < R, t∈(−α, α)}
meeting “bottom” and “top”. We use this fact to produce a transboundary mass distribution forMU,L((∂iA, ∂oA;U ))that has small mass ifαis small.
We use the flat metric onC∗as base metric and set
ρ(u)=1/(Q1) foru∈M∩U,
andρ=0 elsewhere onU, where
U=U\(Q2∪ · · · ∪Qn−1)=A\(Q1∪ · · · ∪Qn−1). Moreover, fori∈ {2, . . . , n−1}we set
ρi=(Qi)/(Q1) ifQi∩M= ∅
andρi=0 otherwise. By considerations very similar to the ones in the proof of Proposition11.2one can show that this transboundary mass distribution is admissible for.
A C∗-square Q that meets M and is disjoint from Q1 must satisfy (Q) <2α. This implies Q⊆M:= {seit :re−2α< s < Re2α,−α < t < α}. Hence U ρ2dAC∗+ n−1 i=2 ρi2≤ 1 (Q1)2 AC∗(M∩U)+ Qi∩M=∅ AC∗(Qi) ≤ 1 (Q1)2 AC∗(M) =α(π+α) (π−α)2, and so 0< C(s, k, d)≤MU,L()≤ α(π+α) (π−α)2.
This shows thatα≥c(s, k, d) >0 as desired.
Proposition 11.5 There exists a numberN ∈N, and a functionψ:(0,∞)→ (0,∞)with
lim
t→∞ψ (t )=0
satisfying the following property: ifK= {Qi:i∈I}is a collection of pair- wise disjoint C∗-squares Qi ⊆C∗, and if E and F are arbitrary disjoint
continua inC∗\ i∈Iint(Qi) withC∗(E, F )≥12, then there exists a set
I0⊆I with #I0≤N such that for the transboundary modulus of the path
family(E, F;)in the region=C∗\ i∈I
0Qi with respect to the col- lectionK= {Qi:i∈I\I0}we have
M,K((E, F;))≤ψ (C∗(E, F )).
HereC∗(E, F )denotes (in accordance with our convention from Sect.2) the relative distance ofE andF with respect to the flat metric dC∗ on C∗. Note that ifEandF are as in the statement, then
E, F ⊆C∗\ i∈I int(Qi)⊆C∗\ i∈I0 int(Qi)=.
Proof The proposition immediately follows from Remark 8.8. We have to check the relevant conditions in this remark. For the mass bounds in the metric measure space(C∗, dC∗, AC∗)note that ifa∈C∗, then we have
for allr >0, and
AC∗(BC∗(a, r))=π r2
for allr≤π. The last equality implies that
AC∗(BC∗(a, r))≥π
5r
2
for allr≤sup{diamC∗(Q):Qis aC∗-square} =π√5. So we get the relevant upper and lower mass bounds.
Moreover, it is clear that aC∗-squareQin(C∗, dC∗, AC∗)isμ-fat for some universal constantμ >0. To produce an explicit (non-sharp) constantμlet
x∈Qand 0< r ≤diamC∗(Q)≤√2(Q)be arbitrary. If 0≤s≤(Q)/2, thenQ∩BC∗(x, s)contains at least a “quarter” of the diskBC∗(x, s). If we apply this fors=r/(2√2)≤(Q)/2≤π, we obtain
AC∗(Q∩BC∗(x, r))≥AC∗(Q∩BC∗(x, s))≥ 1 4AC∗(BC∗(x, s)) =π 4s 2= π 32r 2≥ 1 32AC∗(BC∗(x, r)). So we can takeμ=1/32.
Proposition 11.6 In Corollary 9.13 suppose in addition that the Jordan