数理解析研究所講究録 1293 巻 2002 年 119-131

119
Types of complete infinitely sheeted planes
井中

名工大 ( 名誉教授 )留 ( Mitsuru Nakai )三Nagoya Institute of Technology ( Professor Emeritus )1. Introduction

The purpose of this paper is to show the existence of aone parameter familysuch that $W[\alpha]\in O_{G}$ for small $\alpha\in A$ andof complete infinitely sheeted planes.
for largeof is atriple
Recall that acovering (Riemann) surface $X$ , or more preciely,of two Riemann surfaces $X$ and.Y and an analytic mapping of $X$ to Y. It is said to
has aclosed parametric disc $K$ about such thatbe complete (cf. [2]) if every
is smooth, i.e. there is no brancheach component of $f^{-1}(K)$ is compact. When
, whereis equivalent to the regurarity ofpoint in $X$ , the completeness of
lying over the initial pointis regular if, for any arc on
and any pointof , there always exists acontinuation on $X$ along starting from , i.e. there is an(cf. [2]).
arc on $X$ with its initial point such thator theLet $P$ be the plane, or more precisely, $P$ be the finite complex plane, the Riemanninfinite (i.e. extended) complex plane: $|z|\leq+\infty$ (i.e.
sphere). If the covering surface $(X, P, \pi)$ of satisfies the following two conditions, then$X$ , or $(X, P, \pi)$ , is referred to as amultisheeted plane: the cardinal number cardisis the set of positive integers andaconstantfor every $a\in P$ , where$N$ ;for any cover continuation (Xx, $P,$
) of $(X, P, \pi)$ , the closed set
is of logarithmic capacity zero measured on $X’$ , where acover continuation $(X’, P, \pi’)$ of$(X, P, \pi)$ is acovering surface of $P$ such that $X$ is asubsurface of $X’$ such that $\pi’|X=\pi$ . If
, resp.), then $(X, P, \pi)$(the number, which we call the sheet number of $X$ ,is said to be afinitely (infinitely, resp.) sheeted plane.
What we call our type problem is to judge whether $X\in O_{G}$ or not for agiven multisheeted
plane $(X, P, \pi)$ (cf. [7], [11]), whereis the class of parabolic (i.e. not hyperbolic)Riemann surfaces characterized by the nonexistence of Green function on them (cf. e.g.
$[8],[9])$ . If $(X, P, \pi)$ is finitely sheeted plane, then we can easily see that $X\in O_{G}$ alongwith the parabolicity or the ellipticity of . In view of this, we will consider only infinitely$(W[\alpha])_{\alpha\in A}$$W[\alpha]$$W[\alpha]\not\in O_{G}$$\alpha\not\in A$$(X, \mathrm{Y}, f)$$\mathrm{Y}$$f$$a\in \mathrm{Y}$$a$$(X, \mathrm{Y}, f)$$(X, \mathrm{Y}, f)$$(X, \mathrm{Y}, f)$$(X, \mathrm{Y}, f)$$a$$\tilde{a}\in X$$\mathrm{Y}$$\gamma$$\tilde{a}$$\tilde{\gamma}$$\gamma$$\gamma$$f(\tilde{\gamma})=\gamma$$\tilde{a}$$\tilde{\gamma}$$\mathrm{C}:|z|<\infty$$\hat{\mathrm{C}}=\mathrm{C}\cup\{\infty\}$$\hat{\mathrm{C}}$$P$$\pi^{-1}(a)$$\mu_{X}\in \mathrm{N}\cup\{\aleph_{0}\}$$\mathrm{N}$$X’\backslash X$$\pi’$$\aleph_{0}=\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}$$\mu_{X}\in \mathrm{N}$$O_{G}$$P$$\mu_{X}=\aleph_{0}$120sheeted planes in our type problem. We have afeeling that complete infinitely sheetedofplanes are apt to become parabolic. In fact, we have shown the existence of afamily
which is an equivalence class of infinitely sheeted planes
infinitely sheeted planes
andare similar to each other ifwith respect to the similarity, where
onto
there is atopological mapping ofwith the property that $p\in X_{1}$ is abranch
point of multiplicity $m$ if and only if $T(p)\in X_{2}$ is abranch point of multiplicity $m$ and in
this case $\pi_{1}(p)=\pi_{2}(T(p))$ , such that $X\in O_{G}$ for a
if and only ifiscomplete (cf. [5] and [6]). Nevertheless, the main purpose of this paper is to maintain thatthe parabolicity and the completenes of infinitely sheeted planes have absolutely nothing
to do with each other. This will be shown by constructing aone sequential parameterfamilyof complete infinitely sheeted planes
such that$A$$W[\alpha]\in O_{G}$ if
if ais “large”, where is the family of sequences
is “small” andwith $0<a_{n}\leq 1/2$ . The one sequential parameterfamily
plays the same role as that ofof.
$\mathcal{W}$$(X, \mathrm{C}, \pi)$$(X_{2}, \mathrm{C}, \pi_{2})$$(X_{1}, \mathrm{C}, \pi_{1})$$T$$X_{2}$$X_{1}$$(X, \mathrm{C}, \pi)\in \mathcal{W}$$(X, \mathrm{C}, \pi)$$(W[\alpha],\hat{\mathrm{C}}, \pi)(\alpha\in A)$$(W[\alpha])_{\alpha\in A}$$W[\alpha]\not\in O_{G}$$\alpha$$\alpha=(a_{n})_{n\in \mathrm{N}}$$(W[\alpha]\backslash \pi^{-1}(\infty))_{\alpha\in A}$$(W[\alpha]\backslash \pi^{-1}(\infty), \mathrm{C}, \pi)(\alpha\in A)$$(W[\alpha])_{\alpha\in A}$

2. Construction of aone parameter familyof real numbers
Consider the class $A$ of sequenceswith $0<a_{n}\leq 1/2(n\in$. We view $A=\{\alpha\}$ the sequential parameters family. We will construct acomplete
infinitely sheeted plane
for each $\alpha\in A$ so that we obtain aone sequentialof complete infinitely sheeted planes.
parameter family
Let $J=[0, 1]$ be the slit in C. We denote by the surface obtained from two copies ofby pasting
andcrosswise along . The closure of one ofin will. The other closure of
be called the upper sheet of and denoted byleft in isthen referred to as the lower sheet of and denoted by
The surface is conformallythe sphere but presently represented as atw0-sheeted plane.
In addition to $A$ and we use the index set $\Lambda=\{-2, -1,1, 2\}$ as another ingredientin our construction of $W[\alpha](\alpha\in A)$ . An admissible -tupleof elements in, in Asuch that $i_{k}\neq-i_{k-1}$ $(k=2, \cdots, n)$ . There areis an ordered -tuple of , i2,$4=4\cdot 3^{1-1}$
admissible 1-tuples in , 4 $3=4\cdot 3^{2-1}$ admissible 2-tuples, and in general 4.
, and so on,
admissible 3-tuples
admissible -tuples.
First consider 4slits $I_{0i}=[|i|, |i|+1/2](i\in\Lambda)$ in by the following rule:for$i>0$ andare infor $i<0$ so thatandand lower andare inThen we consider the 4-slitted :$\alpha=(a_{n})_{n\in \mathrm{N}}$$a_{n}$$\mathrm{N})$$(W[\alpha],\hat{\mathrm{C}}, \pi)$$W[\alpha]$$(W[\alpha])_{\alpha\in A}$$S$$\hat{\mathrm{C}}\backslash J$$J$$\hat{\mathrm{C}}\backslash J$$\hat{\mathrm{C}}\backslash J$$S$$\hat{\mathrm{C}}\backslash J$$S^{+}$$\hat{\mathrm{C}}\backslash J$

$S$

$S$

$S$

$S$

$S^{-}$$\hat{\mathrm{C}}$

$S$

$n$

$n$

$i_{1}$$\cdots$$i_{1}i_{2}\cdots$$\Lambda$$i_{n}$$i_{n}$$i_{1}$$i_{1}i_{2},4\cdot 3^{2}=4\cdot 3^{3-1}$$\Lambda$$\cdot$$3^{n-1}$$\cdots$$i_{1}i_{2}i_{3}$$S$$I_{0i}\subset S^{-}$$I_{01}$$I_{02}$$i_{1}\cdots i_{n}$$n$$I_{0i}\subset S^{+}$$S^{+}$$\mathrm{J}\mathrm{o}_{-},2$$S^{-}$$S$(1)

We now fix an$F_{0}:=S\backslash \cup I_{0i}i\in\Lambda^{\cdot}$$\alpha=(a_{n})_{n\in \mathrm{N}}\in A$

so that $0<a_{n}\leq 1/2(n\in \mathrm{N})$ . Let$I_{ni}:=\mathrm{t}\mathrm{t}\mathrm{i}$) $|i|+a_{n}$ ]

121

(2)

$F_{i_{1}\cdot i_{n}}:=S\backslash (I_{n-1,i_{n}}\cup(_{i\in\Lambda\backslash i_{n}}\cup I_{ni}))$for eachsheeted planes$i_{1}\cdots i_{n}$

(3)

of 4

$W[\alpha]$. We then form an infinitely
admissible -tuples in, which we express symbolically as follows:$\cdot 3^{n-1}$$\Lambda(n\in \mathrm{N})$$n$$W[ \alpha]:=F_{0}+\sum_{i_{1}}F_{i_{1}}+\sum_{i_{1}i_{2}}F_{i_{1}i_{2}}+\cdots+\sum_{i_{1}\cdots i_{n}}F_{i_{1}\cdots i_{n}}+\cdots$

,

where
are admissible -tuples. Here we add afew words to mention theexact procedure in the construction in (3). As the first step, we perform the following: each
. Promof the four
in
in
and
is pasted tocrosswise along$n^{th}step$, we perform the following: eachthe second step on, and in general as thefor everyof 4of threeis pasted to each
in
in
and
admissible $(n-1)$ -tuple
in crosswise along.
From the construction it is clear that each $p\in W[\alpha]$ lies above apoint in , which wedenote by $\pi(p)$ . Then $p\mapsto\pi(p)$ is an analytic mapping of
to C. Now it is easy to see, or more preciselythat, is acomplete infinitely sheeted plane. Moreover
constructed
is planar. Concerning the one parameter family
have the following result.
$i_{1}\cdots$$(n\in \mathrm{N})$$n$$i_{n}$$F_{i_{1}}(i_{1}\in\Lambda)$$F_{i_{1}}$$F_{0}$$I_{0,-i_{1}}$$F_{0}$$F_{i_{1}}$$I_{0i_{1}}$$F_{i_{1}\cdots i_{n-1}i_{n}}$$F_{i_{1}\cdots i_{n-1}i_{n}}(i_{n}\in\Lambda\backslash \{-i_{n-1}\})$$i_{1}\cdots i_{n-1}$$F_{i_{1}\cdots i_{n-1}}$$\cdot 3^{n-2}F_{i_{1}\cdots i_{n-1}}$$I_{n-1,-i_{n}}$$\Lambda$$I_{n-1,i_{n}}$$F_{i_{1}\cdots i_{n-1}}$$F_{i_{1}\cdots i_{n-1}i_{n}}$$\hat{\mathrm{C}}$$W[\alpha]$$W[\alpha]$$(W[\alpha],\hat{\mathrm{C}}, \pi)$$W[\alpha]$$(W[\alpha])_{\alpha\in A}$$\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{v}\mathrm{e},\mathrm{w}\mathrm{e}$deter mined by the sequence
THEOREM. The complete infinitely sheeted plane$i.e$, if is large in the.
with $0<a_{n}\leq 1/2(n\in \mathrm{N})$ is hyperbolic,sense $e.g$ . that$\alpha=$$W[\alpha]$$W[\alpha]\not\in O_{G}$$(a_{n})_{n\in \mathrm{N}}$

(4)

and
(5)

$\inf_{n\in \mathrm{N}}a_{n}>0$$W[\alpha]$

is parabolic, $i.e$ . $W[\alpha]\in O_{G}$ , if$\alpha$$\alpha$;is small in the sense $e.g$ . that$\sum_{n\in \mathrm{N}}\frac{1}{3^{n}}\log\frac{1}{a_{n}}=+\infty$.

The proof of the second part of our thorem above will be givn in the next \S 3 and thefirst part in \S 4. The proof of the second part is easy and very short. Since the proof given
in \S 4 is considerably long, the section is divided into 3subsections \S \S 4.1-4.3.

3. Proof of Theorem: parabolicity

We now start the proof of our theorem stated in \S 2. In this present short \S 3 we prove that

122
if

$\alpha=(a_{n})_{n\in \mathrm{N}}\in A$$W:=W[\alpha]$ .

(6)

Let

$W_{n}$$W_{0}:=F_{0}$is so small as to satisfy (5), then $W[\alpha]\in O_{G}$ . For simplicity we set(nbe regular subregions of W given by$\geq 0)$,

$W_{n}:=F_{0}+ \sum_{i_{1}}F_{i_{1}}+\sum_{i_{1}i_{2}}F_{i_{1}i_{2}}+\cdots+\sum_{i_{1}i_{n}}F_{i_{1}}$. $i_{n}(n\in \mathrm{N})$ ,where $i_{1}\cdots i_{k}(1\leq k\leq n)$ are admissible -tuples $(1 \leq k\leq n)$ . Then
forms
$W$
$W$
in this present
aregular exhaustion of , which is called the standard exhaustion of
be the continuous function onsuch thatletpaper. For each
and $w_{n}|\partial W_{n-1}=0$ . The
with boundary values
is harmonic onon. The Dirichletis referred to as the harmonic measure offunctionis, by definition,ofoverintegral$k$$n\in \mathrm{N}$$(W_{n})_{n\geq 0}$$\overline{W_{n}}\backslash W_{n-1}$$w_{n}$$w_{n}|\partial W_{n}=1$$W_{n}\backslash \overline{W_{n-1}}$$W_{n}\backslash \overline{W_{n-1}}$$\partial W_{n}$$w_{n}$$D_{W_{n}\backslash \overline{W_{n-1}}}(w_{n})$$w_{n}$$W_{n}\backslash \overline{W_{n-1}}$$D_{W_{n}\backslash \overline{W_{n-1}}}(w_{n}):= \int_{W_{n}\backslash \overline{W_{n-1}}}|\nabla w_{n}(z)|^{2}dxdy(z=x+\sqrt{-1}y)$Then the modulus$\mu_{n}$$w_{n}$of the configulation$W_{n}\backslash \overline{W_{n-1}}$.

is given by$\mu_{n}:=2\pi/D_{W_{n}\backslash \overline{W_{n-1}}}(w_{n})$.

By virtue of (5) we will derive(7)$\sum_{n\in \mathrm{N}}\mu_{n}=+\infty$.

Before proceeding to the proof of the above (7), we pause here to make the following
simple observation. We denote by
for $0<a<1/4$ the annulus bounded by the ellipse
centered at the origin 0with major axis 1and minor axis $\sqrt{1-4a^{2}}$ and the segment $[-a, a]$ ,by
the annulus bounded by the ellipse centered at 0with major axis $1/a$ and minor axis$\sqrt{1-4a^{2}}/a$ and the segment [-1, 1], and finally by
the ring domain bounded by circles$|z|=2a/(1+\sqrt{1-4a^{2}})$ and $|z|=1$ . By the conformal mapping $w=(z+1/z)/2$ ,is. In view of. By the conformal mapping $w=az$ ,mapped onto
is mapped onto
, where e.g.
means the modulus of , and$A_{a}$$B_{a}$$C_{a}$$C_{a}$$\mathrm{m}\mathrm{o}\mathrm{d} A_{a}=\mathrm{m}\mathrm{o}\mathrm{d}$$A_{a}$$B_{a}$$B_{a}$$B_{a}=\mathrm{m}\mathrm{o}\mathrm{d} C_{a}$$A_{a}$$\mathrm{m}\mathrm{o}\mathrm{d} A_{a}$$\mathrm{m}\mathrm{o}\mathrm{d} C_{a}=\log[(1+\sqrt{1-4a^{2}})/2a]$,

we see that(8)$\mathrm{m}\mathrm{o}\mathrm{d} A_{a}=\log[(1+\sqrt{1-4a^{2}})/2a]=2\mathrm{n}/\mathrm{D}\mathrm{A}\mathrm{n}(\mathrm{w})$whereis the harmonic measure of the segment $[-a, a]$ on
can be derived that
$w$

(9)

$\frac{1}{2}\log\frac{1}{a}\leq \mathrm{m}\mathrm{o}\mathrm{d} A_{a}\leq\log\frac{1}{a}$.

$\partial A_{a}$,. From the above (8) it

123
We now turn to the proof of (7). Observe thatconsists of 4
offor all admissible -tuples. Since the modulus$2\pi/D_{F_{l}1}i_{n}(w_{n})$ , whereis the harmonic measure ofon$W_{n}\backslash \overline{W_{n-1}}$$F_{i_{1}}i_{n}$$i_{1}\cdots i_{n}$$n$$\mu_{i_{1}}i_{n}$$F_{i_{1}}i_{n}$$W_{n}\backslash \overline{W_{n-1}}$$\partial W_{n}$$w_{n}$$\cdot 3^{n-1}$$D_{W_{n}\backslash \overline{W_{n-1}}}(w_{n})= \sum_{i_{1}\cdot i_{n}}D_{i_{1}\cdot\cdot i_{n}}(w_{n})$components
is given by

, and

,

we conclude that
(10)

$1/ \mu_{n}=\sum_{i_{1}\cdots i_{n}}1/\mu_{i_{1}\cdots i_{n}}$.

Let $A:=A_{a_{n}-1/2}+(|i_{n}|+a_{n-1}/2)$ , where $a_{0}=1/2$ , and be the harmonic measure of
the segment part of, the Dirichlet principle implies
on . Since$w$$\partial A$$A$$A\subset F_{i_{1}\cdot\cdot i_{n}}$$D_{A}(w)=D_{F_{i_{1}}}$

where

$w$

in

$(w)\geq D_{F_{i_{1}}}i_{n}(w_{n})$is extended tofrom by setting $w=0$ on. Using (10) we see that$A$

$F_{i_{1}\cdot\cdot i_{n}}$,$F_{i_{1}\cdots i_{n}}\backslash A$. Thus$\mathrm{m}\mathrm{o}\mathrm{d} A_{a_{n-1}/2}=$$\mathrm{m}\mathrm{o}\mathrm{d} A\leq\mu_{i_{1}\cdots i_{n}}$$1/ \mu_{n}=\sum_{i_{1}\cdot\cdot i_{n}}1/(\frac{1}{2}\log\frac{1}{a_{n-1}})=4\cdot 3^{n-1}/(\frac{1}{2}\log\frac{1}{a_{n-1}})$or $\mu_{n}\geq 8^{-1}3^{-(n-1)}\log(1/a_{n-1})$ . Therefore (5) implies$\sum_{n\in \mathrm{N}}\mu_{n}\geq 8^{-1}\sum_{n\in \mathrm{N}}3^{-(n-1)}\log(1/a_{n-1})=+\infty$,

which is the relation (7) to be proved.
By the sa -Noshiro modular criterion for the parabolicity of $W$ (cf. e.g. $[8],[9]$ , etc.),
the existence of the regular exhaustion, which we call the standard exhaustion of$W$ in this paper, of $W$ with (7)concludes that $W\in O_{G}$ .
$\mathrm{r}\mathrm{i}\mathrm{o}$$(W_{n})_{n\in \mathrm{N}}$$\square$

4. Proof of Theorem: hyperbolicity
Under the assumption that the sequenceis large in the sense that $c:=$, we will show that
. The largest menberis hyperbolic, i.e.
in $A$ may be said to be the sequencesuch that $s_{n}=1/2(n\in \mathrm{N})$ so that$\alpha=(a_{n})_{n\in \mathrm{N}}$$\inf_{n\in \mathrm{N}}a_{n}>0$$W[\alpha]\not\in O_{G}$$W[\alpha]$$\sigma:=(s_{n})_{\mathrm{N}\in \mathrm{N}}$$\sigma=(1/2,1/2, \cdots, 1/2, \cdots)$.

The plan of our proof forgoes as follows. First we will show in Subsection 4.1
the existence of aquasiconformal mapping of. By the quasiconformal
ontoinvariance of the parabolicity (i.e. if there is aquasiconformal mapping of aRiemannsurface, then $R_{1}\in O_{G}$ if and oly if $R_{2}\in O_{G}$ (cf. e.g. [9]) , we canonto anotherconclude our objective
if we show that
. Then, as the second step$\mathrm{W}[\mathrm{a}]\not\in O_{G}$$T$$R_{1}$$W[\sigma]$$W[\alpha]$$R_{2}$$W[\alpha]\not\in O_{G}$$)$$\mathrm{W}[\mathrm{a}]\not\in O_{G}$124
we will derive in Subsections 4.2 and 4.3 acontradiction from the errorneous assumption$W[\sigma]\in O_{G}$.

4.1. Construction of aquasiconformal mapping. In this first Subsection 4.1, we
will construct aquasiconformal mapping of. We consider, as we didonto
in Q3, the standard exhaustions for covering surfaces constructed in \S 2. Recall that thestandard exhaustionofis given by$T$,$W[\alpha]$$W[\alpha]$$(W_{n}[\alpha])_{n\geq 0}$$W_{0}[\alpha]:=F_{0}$$W[\sigma]$$W_{n}[ \alpha]:=F_{0}+\sum_{i_{1}}F_{i_{1}}+\cdots+\sum_{i_{1}\cdots i_{n}}F_{i_{1}\cdots i_{n}}(n\geq 1)$

,

where
will also be denoted byindicating the dependence on . Take thestandard neighborhoodof eachcharacterized by the following three
conditions: a) each component of
is apiecewise smooth annulus containing only one, where aregion is piecewise smooth if the region is relatively compact
component ofand its relative boundary consists of afinite number of mutually disjoint piecewise smooth
Jordan curves; b) any two different components ofare disjoint in their closures;c)is an open set independent of so that, for example,$F_{i_{1}\cdots i_{k}}[\alpha]$$F_{i_{1}\cdots i_{k}}$$\alpha$$\partial W_{n}[\alpha](n\in \mathrm{N})$$U_{n}[\alpha]$$U_{n}[\alpha]$$\partial W_{n}[\alpha]$$\bigcup_{n\geq 0}U_{n}[\alpha]$$W[ \alpha]\backslash \bigcup_{n\geq 0}\overline{U_{n}[\alpha]}$$\alpha$$W[ \alpha]\backslash \bigcup_{n\geq 0}\overline{U_{n}[\alpha]}=W[\sigma]\backslash \bigcup_{n\geq 0}\overline{U_{n}[\sigma]}$.

Let
be aJordan curve which is acomponent ofand letbe the annulus
containing
acomponent
which isof.
The construction of goes as follows. First let the mapping be defined as the identity
betweenand. The construction will be over if we$K$
define a -quasiconformal mapping of
onto
for every
and its corresponding:
such thatis the identity and $K\leq K_{0}$ , whereis aconstant
independent of the choice of(and).
Supposecomes from the slit $I_{n+1,i_{n+1}}=I_{n+1,i_{n+1}}[\alpha]=[|i_{n+1}|, |i_{n+1}|+a_{n}]$ inand the slit $I_{n,-i_{n+1}}=I_{n,-i_{n+1}}[\alpha]$ in, which are pasted
crosswise and give rise to the analytic Jordan curve. Let
be the corresponding
one toinso that it comes from $I_{n+1,i_{n+1}}[\sigma]=[|i_{n+1}|, |i_{n+1}|+s_{n}]$ in
and $I_{n,-i_{n+1}}[\sigma]=[|-i_{n+1}|, |-i_{n+1}|+s_{n}]$ in, which are pasted crosswise and giverise to.
We denote by $Z(d)$ the slitted square$\gamma[\alpha]$$V[\alpha]$$\partial W_{n}[\alpha]$$U_{n}[\alpha]$$\gamma[\alpha]$$T$$W[ \sigma]\backslash \bigcup_{n\geq 0}U_{n}[\sigma]$$T$$W[ \alpha]\backslash \bigcup_{n\geq 0}U_{n}[\alpha]$$V[\sigma]$$T$$V[\sigma]$$V[\alpha]$$V[\alpha]$$\partial V[\sigma]\mapsto\partial V[\alpha]$$V[\alpha]$$K_{0}$$V[\sigma]$$\gamma[\alpha]$$F_{i_{1}\cdots i_{n}i_{n+1}}$$=F_{i_{1}\cdots i_{n}i_{n+1}}[\alpha]$$F_{i_{1}\cdots i_{n}}=F_{i_{1}\cdots i_{n}}[\alpha]$$\gamma[\alpha]$$\gamma[\alpha]$$W[\sigma]$$\gamma[\sigma]$$F_{i_{1}\cdots i_{n}i_{n+1}}[\sigma]$$F_{i_{1}\cdots i_{n}}[\sigma]$$\gamma[\sigma]$$Z(d):=Q\backslash J(d)$,where $Q=$ $(0, 1)$
is the unit square and $J(d)=[1/4,1/4+d]+\sqrt{-1}/2$ is the line
segment of the length $0<d\leq 1/2$ and set$\cross(0, 1)$$Z_{i}(d):=Z(d)-\sqrt{-1}/2+|i|$125
and the slit ofis viewed as being contained inThen
is viewed as being contained in. Similarly
is identical with
we take the. Thus as
is identical withand the slit of.
andcrosswise alongtoannulus obtained by pastingto
If we can construct aquasiconfo rmal mapping ofsuch that is identity on their boundaries andfor$i\in\Lambda$.

$F_{i_{1}}i_{n}i_{n+1}[\alpha]$$Z_{i_{n+1}}(a_{n})$$Z_{-i_{n+1}}(a_{n})$$I_{n+1,i_{n+1}}[\alpha]$$Z_{i_{n+1}}(a_{n})$$T$$I_{n,-i_{n+1}}$$I_{n+1,i_{n+1}}$$Z_{-i_{n+1}}(a_{n})$$Z_{i_{n+1}}(a_{n})$$I_{n+1,i_{n+1}}[\alpha]$$V[\alpha]$$I_{n,-i_{n+1}}[\alpha]$$Z_{-i_{n+1}}(a_{n})$$F_{i_{1}}i_{n}[\alpha]$$Z_{i_{n+1}}(s_{n})\cup I_{n+1,i_{n+1}}[\sigma]$$Z_{i_{n+1}}(a_{n})\cup$$T$$T(I_{n+1,i_{n+1}}[\sigma])=I_{n+1,i_{n+1}}[\alpha]$,with the dilatation
tothen it can be continued symmetrically to the mapping of.
andindependent ofless than afixed constantBased upon the observation above we, hence, only have to show the existence of asuch that
quaiconformal mapping from the square $Q=(0,1)\cross(0, 1)$ to itself fixing$w(I(1/2))=I(a_{n})$ , where$T$$K_{0}$$V[\sigma]$$V[\alpha]$$\gamma[\alpha]$$\gamma[\sigma]$$\partial Q$$w$$I(1/2):=[1/4,1/4+1/2]\sqrt{-1}/2$ ,$I(a_{n}):=[1/4,1/4+a_{n}]+\sqrt{-1}/2$

depending only on . For simplicity weand the dilatation $K$ is bounded by aconstant
set $a_{n}=:d$ in this proof so that $I(d):=I(a_{n})$ and $0<c\leq d\leq 1/2$ .
Consider ahomeomorphic mapping of onto itself given by$K_{0}$

$w$

$c$

$Q$

$w(z):=u(x)+\sqrt{-1}y(z=x+\sqrt{-1}y)$ ,where $u(z):=x$ for $0\leq x\leq 1/4,2(\xi(y)-1/4)(x-1/4)+1/4$ for $1/4\leq x\leq 2/4$ , and$4(1-\xi(y)(x-1)+1$ for $3/4\leq x\leq 1$ , where$\xi(y):=(1-2d)|y-1/2|+(1/4+d)$ ,which maps $[0, 1]$ to $[1/4+d, 3/4]$ . Then $w_{x}=u_{x}$ and $w_{y}=u_{y}+\sqrt{-1}$ and thus $w_{z}=$$(w_{x}-\sqrt{-1}w_{y})/2=(u_{x}+1-\sqrt{-1}u_{y})/2$ and $w_{\overline{z}}=(w_{x}+\sqrt{-1}w_{y})/2=(u_{x}-1+\sqrt{-1}u_{y})/2$
so that the candidate of complex dilatation$\mu(z):=w_{\overline{z}}(z)/w_{z}(z)$satisfies $|\mu|^{2}=(P-p)/(P+p)$ , where $P:=(u_{x}^{2}+u_{y}^{2}+1)/2$ and $p:=u_{x}$ . Since $P\leq P_{0}:=$$11/2$ and $p\geq p_{0}:=2d$ , we have$| \mu|^{2}=\frac{P-p}{P+p}\leq\frac{P_{0}-p_{0}}{P_{0}+p_{0}}=(\frac{K-1}{K+1})^{2}$or equivalently$| \mu|\leq\frac{K-1}{K+1}$

,

where
$K:= \frac{11}{4d}+\sqrt{\frac{121}{16d^{2}}-1}\leq\frac{11}{4c}+\sqrt{\frac{121}{16c^{2}}-1}=:K_{0}$126$K \geq\frac{11}{4\cdot(1/2)}+\sqrt\frac{121}{16\cdot(1/2)^{2}}-1=\frac{11}{2}+\sqrt{\frac{1}{4}+29}>1$.

such that $|\mu|\leq(K-1)/(K+1)$
Hence defined above is ameasurable function on:
so that is eligible to be acomplex dilatation on . Clearlyon
is a, ACL onhomeomorphism, the identity on
and $w(I(1/2))=I(d)$ , and satisfies theBeltrami equation$\overline{Q}$

$\mu$

$w$

$\overline{Q}$$\overline{Q}$$\mu$$\overline{Q}\mapsto\overline{Q}$$\overline{Q}$$\partial Q$$w_{\overline{z}}(z)=\mu(z)w_{z}(z)$almost everywhere on , i.e.
is ageneralized solution of the above Beltrami equation on$K$. Therefore is arequired -quasiconformal mapping of to (cf. e.g. [3]) fixing
pointwise with $K\leq K_{0}$ , afixed constant depending only upon .
$w$$\overline{Q}$$w$$\overline{Q}$$\overline{Q}$$\overline{Q}$$\partial Q$$c$4.2. Exhausting regions with short boundaries. We are in the position to prove. We will do this by contradiction so that we now make the errorneous assumption that $W:=W[\sigma]\in O_{G}$ . Then there exists an Evans-Selberg potential on $W$(cf. [4], see also [9] and [10]):
is aharmonic function onsuch that has the$a\inW$$z\mapsto
h(z)+\log(1/|z|)$, i.e.
negative logrithmic pole atis harmonic at for the local
parameter at with $z(a)=0$ , and $h(z)arrow+\infty$ as tends to the Alexandroff point of$W$ . We consider the polar coordinate
on $W$ constructed from , where $r=e^{h}$ and0is the multivalued conjugate harmonic function of onso that $d\theta=*dh$ on.
Using abranch of 0we can use
as local paeameters at each point of $W$ except for
an isolated point set of points with $|\nabla h|=0$ .
For each $0<t<+\infty$ we set$W[\sigma]\not\in O_{G}$

$h$

$h$

$h$

$W\backslash a$

$a$

$z$

$a$

$z$

$re^{\sqrt{-1}\theta}$

$h$

$h$

$W\backslash a$$W\backslash a$$re^{\sqrt{-1}\theta}$$\theta$$R(t):=\{z\in W : h(z)<\log t\}=\{z\in W : r=r(z)<t\}$so that $a\in R(t)$ since $r(a)=0$ . Clearly $R(t)$ is arelatively compact region and its relative
boundary $\Gamma(t):=\partial R(t)$ consists of afinite number of Jordan curves which may havecommon points at. Except these points which are isolated in $W$ the above
Jordan curves are analytic. Hence except for acountable number of $0<t<+\infty$ , $R(t)$ is) consists of afinite number of mutually disjoint analytic Jordan
aregular region andcurves. Clerly.
$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}*dh=0$$\Gamma\langle t$$\int_{\Gamma(t)}d\theta=2\pi$

In the covering surface
we consider the induced spherical metric on $W$ induced
from by the projection . The area of aset $G\subset W$ is denoted by $|G|$ and the length ofacurveis also denoted byin the above induced metric. Put $A(t):=|R(t)|$ , i.e.
the area of $R(t)$ . By denoting $\pi’(z)=d\pi(z)/dz$ , we have$(W,\hat{\mathrm{C}}, \pi)$$\hat{\mathrm{C}}$$\pi$$\gamma\subset W$$|\gamma|$$A(t):= \int\int_{R(t)}(\frac{|\pi’(z)|}{1+|\pi(z)|^{2}})^{2}rdrd\theta$

127$= \int_{0}^{t}(\int_{\Gamma(r)}(\frac{|\pi’(z)|}{1+|\pi(z)|^{2}})^{2}rd\theta)dr(z=re^{\sqrt{-1}\theta})$.

Clearly $t\mapsto A(t)$ is continuous and$A( \infty)=\int\int_{W}(\frac{|\pi’(z)|}{1+|\pi(z)|^{2}})^{2}$rdrr$\theta=\mu w\int\int_{F_{0}}(\frac{|\pi’(z)|}{1+|\pi(z)|^{2}})^{2}rdrd\theta=+\infty$.

Hence we conclude that
(11)

$A(\infty)=1\mathrm{m}A(t)t\mathrm{j}\infty=+\infty$

Put $L(t):=|\Gamma(t)|$ , the length of$\Gamma(t)=\mathrm{L}(\mathrm{t})$.

, so that$L(t):= \int_{\Gamma(t)}\frac{|\pi’(z)|}{1+|\pi(z)|^{2}}td\theta$$(z=re^{\sqrt{-1}\theta})$,which is also continuous in . Denoting by $A’(t)=dA(t)/dt$ , we have$t$$A’(t)= \int_{\Gamma(t)}(\frac{|\pi’(z)|}{1+|\pi(z)|^{2}})^{2}td\theta$.

By th Schwarz inequality we see that
$L(t)^{2}=( \int_{\Gamma(t)}\frac{|\pi’(z)|}{1+|\pi(z)|^{2}}td\theta)^{2}$$\leq(\int_{\Gamma(t)}(\frac{|\pi’(z)|}{1+|\pi(z)|^{2}})^{2}td\theta)\cdot(\int_{\Gamma(t)}td\theta)=2\pi tA’(t)$and thus we have obtained the inequality(12)$L(t)^{2}\leq 2\pi tA’(t)$.

Choose and then fix an arbitrary $1/2<\tau<1$ . We consider–(-S) $:=\{t\in[s, \infty) : L(t)\geq A(t)^{\tau}\}$ ,. Choose an $r_{0}>0$ such that $R(r_{0})$ is aregular region. Take an$s_{n} \in(\max\{n, r_{n-1}\}, \infty)$ for each
. Considering the logarithmic length

which is closed in$[5, \infty)$$n\in \mathrm{N}$$\mathcal{L}(X):=\int_{X}\frac{dt}{t}$for measurable subset $X\subset(0, \infty)$ , we observe that$\mathcal{L}([s_{n}, \infty))=\int_{s_{n}}^{\infty}\frac{dt}{t}=[\log t]_{s_{n}}^{\infty}=+\infty$.

128
Using $1/t\leq 2\pi A’(t)/L(t)^{2}$ for t$2\pi A’(t)/A(t)^{2\tau}$ for t, we deduce$\in[s_{n}, \infty)$

as aconsequence of (12) and $2\pi A’(t)/L(t)^{2}\leq$$\in\Xi(s_{n})$$\mathcal{L}(_{-}^{-}-(s_{n}))=\int_{-(s_{n})}--\frac{dt}{t}\leq 2\pi\int_{-(s_{n})}--fracA’(t)A(t)^{2\tau}dt$$\leq 2\pi\int_{s_{n}}^{\infty}\frac{A’(t)}{A(t)^{2\tau}}=2\pi\int_{A(s_{n})}^{A(\infty)}\frac{dA}{A^{2\tau}}$$=2 \pi[\frac{1}{1-2\tau}A^{-2\tau+1}]_{A(s_{n})}^{\infty}=\frac{2\pi}{2\tau-1}A(s_{n})^{-2\tau+1}<\infty$.

Hence $([s_{n}, \infty)\backslash ---(s_{n}))=\infty-(2\pi/(2\tau-1))A(s_{n})^{-2\tau+1}=+\infty$ . This shows that the set, i.e.
withis open and afortiori we can find an, such that $R(r_{n})$ is regular. Since $r_{n}>s_{n}> \max\{n, r_{n-1}\}$ , we have$r_{n}\in(s_{n}, \infty)$$[s_{n}, \infty)\backslash ---(s_{n})\neq\emptyset$$r_{n}\not\in---(s_{n})$$L(rn)<\mathrm{A}(\mathrm{r}\mathrm{n})\mathrm{T}$$0<r_{0}<r_{1}<r_{2}<\cdots<r_{n-1}<r_{n}\uparrow\infty$.

Using this sequence we finally define(13)$\Gamma_{n}=\mathrm{L}(\mathrm{r}\mathrm{n})=\partial R(r_{n})$$R_{n}:=\mathrm{H}(\mathrm{s}\mathrm{n}))$Then$(R_{n})_{n\geq 0}$.

$(n\geq 0)$

is aregular exhaustion of $W$ and.

$|\Gamma_{n}|/|R_{n}|=L(r_{n})/A(r_{n})<A(r_{n})^{\tau}/A(r_{n})=A(r_{n})^{\tau-1}\downarrow 0(n\uparrow\infty)$

Thus the boundaries
that$\Gamma_{n}$of the exhausting regions(14)$R_{n}$$\lim_{narrow\infty}\frac{|\Gamma_{n}|}{|R_{n}|}=0$

of $W$ are relatively short in the sense.

Exhausting regions with long boundaries. As aconsequence of $W:=$, we have seen in the preceding subsection 4.2 the existence of aregular exof $W$ given in (13) that the boundaries
haustionofare so short as
to satisfy (14). We will see from adifferent view point that the same exhaustion
satisfies

4.3.

$W[\sigma]\in O_{G}$$(R_{n})_{n\geq 0}$$\Gamma_{n}=\partial R_{n}$$R_{n}$$(R_{n})_{n\geq 0}$(15)$\lim_{narrow}\inf_{\infty}\frac{|\Gamma_{n}|}{|R_{n}|}>0$

as aconsequence of $W\in O_{G)}$ which contradicts (14). Thus we should not have assumed$W\in O_{G}$ and the proof of $W:=W[\sigma]\not\in O_{G}$ will be over. The proof of (15) goes as follows.
Besides the main covering surface
we have two other covering surfaces $(W, F, \pi_{1})$, whereandinvolved inwith the relationis the closed
by pasting crosswise along
surface of genus 2obtained fromandand also alon$(W,\hat{\mathrm{C}}, \pi)$$(F,\hat{\mathrm{C}}, \pi_{2})$$(W,\hat{\mathrm{C}}, \pi)$$F_{0}$$F$$\pi=\pi_{2}\circ\pi_{1}$$I_{01}$$I_{0,-1}$$\mathrm{g}$

129
and
. The metric on
is the one induced from the sperical metric on
by
$W$
$F$
and that onis the one iduced from that on
by , which is identical with the one
induced from the sperical metric on by already considered in the preceding subsection
4.2. The area (length, resp.) of aset (an arc , resp.) is denoted as before by the same
notation $|G|$ ( , resp.) on eithe one of three, $(W, F, \pi_{1})$ , and.
We denote by $\Phi(W)$ ( $\Phi(F)$ , resp.) the set of closed curves on $W$ ( $F$ , resp.) which are
not of homotopy null. We maintain$I_{02}$$F$$I_{0,-2}$$\hat{\mathrm{C}}$$\pi_{2}$$\pi_{1}$$\hat{\mathrm{C}}$

$\pi$

$G$

$\gamma$$(W,\hat{\mathrm{C}}, \pi)$$(F,\hat{\mathrm{C}}, \pi_{2})$$|\gamma|$$\varphi$$a(W):= \inf_{\varphi\in\Phi(W)}|\varphi|>0$.

For aproof observe first that $(W, F, \pi_{1})$ is complete and therefore the monodromy theorem
assurs thatalong with
: $\pi_{1}(\Phi(W)\subset\Phi(F)$ . Since the metric on$W$ may be viewed as
induced from one on $F$ by , we see that
. Hence$\pi_{1}(\varphi)\in\mathrm{F})$$\varphi\in\mathrm{W})$$|\varphi|\geq|\pi_{1}(\varphi)|$$\pi_{1}$,$a(W)= \inf_{\varphi\in\Phi(W)}|\varphi|\geq\inf_{\varphi\in\Phi(W)}|\pi_{1}(\varphi)|=\inf_{\varphi\in\pi_{1}(\Phi(W))}|\varphi|\geq\inf_{\varphi\in\Phi(F)}|\varphi|$by which we only have to prove(17)$a(F):= \inf_{\varphi\in\Phi(F)}|\varphi|>0$

to conclude (16), i.e. $a(W)\geq a(F)>0$ . Fix an arbitrary $p\in F$ and denote by
set ofpassing through and consider$\varphi\in\Phi(F)$$\Phi_{p}(F)$

the

$p$

$f(p):= \inf_{\varphi\in\Phi_{\mathrm{p}}(F)}|\varphi|$.

Then $a(F)= \inf_{p\in F}f(p)$ . We maintain that $f(p)>0$ . Contrary to the assertion assume
that $f(p)=0$ for a $p\in F$ . Choose an open parametric disc $B$ centered at and let
be the distance between andmeasured by the induced spherical metric on $F$ .
Then there is asuch that, which is impossible since $B$ is simply
connected andis not homotop null. Otherwise every
is not contained in
$B$ ,
i.e.
, and afortiori,, which leads tocontradicting the present errorneous assumption $/(\mathrm{p})=0$ . Thus we have seen that $f(p)>0$
for every $p\in F$ . Next choose an arbitrary pair $(p, q)$ of points in $F$ and an arc
connecting and . For any, we have.
andHence $f(q)\leq f(p)+2d_{F}(p, q)$ , which implies the (Lipschitz) continuity of On $F$ , i.e.
$|f(p)-f(q)|\leq 2d_{F}(p, q)$ for every pair $(p, q)$ of points in $F$ , where $d_{F}(p, q)$ is the induced
spherical distance on $F$ . Thus $f\in C(F)$ and $f>0$ on $F$ implies that infir $f>0$ , which
proves (17).
Concerning the standard exhaustion
introduced in the preceding subsection 4.2,
we suppose the relative boundary
consists of afinite numberof components$\Gamma_{ni}(1\leq i\leq\nu_{n})$ :. Observe thatis not homologue null. Otherwise$p$$\epsilon$$F\backslash B$$p$$\varphi_{0}\in\Phi_{p}(F)$$\varphi_{0}\subset B$$\varphi\in\Phi_{p}(F)$$\varphi_{0}$$\varphi\cap(F\backslash B)\neq\emptyset$$f(p)= \inf_{\varphi\in\Phi_{p}(F)}|\varphi|\geq\epsilon$$|\varphi|\geq\epsilon$$\gamma\subset F$

$p$

$q$

$\varphi\in\Phi_{q}(F)$$\gamma\varphi\gamma^{-1}\in\Phi_{p}(F)$$|\gamma\varphi\gamma^{-1}|\leq|\varphi|+2|\gamma|$$f$$(R_{n})_{n\geq 0}$$\partial R_{n}=:\Gamma_{n}$$\Gamma_{n}=\bigcup_{1\leq i\leq\nu_{n}}\Gamma_{ni}$$\nu_{n}$$\Gamma_{ni}$130. Since $h=r_{n}$is the boundary of arelatively closed region, which we denote by
and thus must be aon
, the maximum principle yields
onof
on $W$ , which is clearly acontradiction. Since the homology groupconstant$W$ is isomorphic to the factor groupof the homotopy groupof$W$ with its commutator subgroup, being homotop null implies being homologue$W$
and thus we see that $\Gamma_{ni}\in\Phi(W)(1\leq i\leq\nu_{n})$ so thatnull for any closed curve on,.
$(\Gamma_{ni})$$\Gamma_{ni}$$h\equiv r_{n}$$\partial(\Gamma_{ni})=\Gamma_{ni}$$(\Gamma_{ni})$$h$$\mathcal{H}(W)$$r_{n}$$\mathcal{F}(\mathcal{W})/\mathcal{F}^{(\infty)}(\mathcal{W})$$\mathcal{F}(\mathcal{W})$$\mathcal{F}^{(\infty)}(\mathcal{W})$$| \Gamma_{n}|=\sum_{1\leq i\leq\nu_{n}}\geq\sum_{1\leq i\leq\nu_{n}}a(W)$(18)$\mathrm{i}.\mathrm{e}$$|\Gamma_{n}|\geq\nu_{n}a(W)(n\in \mathrm{N})$.

( $F$ , resp.). As arelatively, resp.) the Euler characteristic of(
We denote by
is aplanar surface bounded by
Jordancompact subsurface of the planar surface $W$ ,and afortiori $\rho(R_{n})=\nu_{n}-2$ . Since the genus $g(F)$ of $F$ is 2, i.e. $g(F)=2$ , wecurvessee that $\rho(F)=2g(F)-2=2$ . Recall the main theorem of the Ahlfors theory of covering:
surfaces (cf. [1], see also e.g. $[7],[10]$ ) applied to the covering surface$\rho(R_{n})$$\rho(F)$$R_{n}$$R_{n}$$\nu_{n}$$\Gamma_{ni}$$(R_{n}, F, \pi_{1})$(19)$\max\{\rho(R_{n}), \mathrm{O}\}\geq\rho(F)\frac{|R_{n}|}{|F|}-h_{F}|\Gamma_{n}|$,. The inis apositive number depending only upon $F$ and independent of
whereequality (18) implies $|\Gamma_{n}|/a(W)\geq\nu_{n}$ and the relation $\rho(R_{n})=\nu_{n}-2$ yields $\nu_{n}>\rho(R_{n})$ .
deuce $|\Gamma_{n}|/a(W)>\rho(R_{n})$ holds. These with $\rho(F)=2$ transform (19) into the followingform:$R_{n}$$h_{F}$$|\Gamma_{n}|/a(W)>2|R_{n}|/|F|-h_{F}|\Gamma_{n}|$.

Rewriting the above inequality with $b:=2a(W)/|F|(a(W)h_{F}+1)>0$ , we obtain theinequality $|\Gamma_{n}|/|R_{n}|>b>0$ for every
and thus we can finally deuce (15):$n\in \mathrm{N}$$\lim\inf_{narrow\infty}|\Gamma_{n}|/|R_{n}|\geq b>0$.

$\square$

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84-93.
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