RIEMANN SURFACES WITH APPLICATIONS TO CIRCULAR.RADIAL SLIT MAPPINGS
BY
A. M A R D E N and B. R O D I N
University of Minnesota, Minneapolis and University of California, San DiegoQ)
m
Partition the b o u n d a r y of a compact bordered R i e m a n n surface W into four disjoint sets ~0, zr fl, y with ~0 and zr non-empty. L e t t~ denote the compactification of W obtained b y adding to W a point for each b o u n d a r y component. Define
A
_~ = {c : c is an arc in W - 7 from ~0 to ~}
and F* = {c :c is a sum of closed curves in l~ z - / ~ such t h a t c separates ~0 from ~r Determine the harmonic function u in W b y the b o u n d a r y conditions u = 0 on ~0, u = 1 on ~, ~u/~n=O along y and u is constant on each component ~ in ~ such t h a t .[~du* =0.
Then 2 ( F ) = ]]du][ -2, 2 ( F * ) = ][du[[ ~ (see L e m m a I I I . l . 1 ) where 2 ( ' ) denotes the extremal length a n d [[du[] ~ the Dirichlet integral. This result was essentially known to Ahlfors and Beurting b y the time of their f u n d a m e n t a l p a p e r on eonformal invariants [I]. We observe t h a t if W is planar and ~0, ~ are each single b o u n d a r y components, exp 2zt(u + iu*)] [[dull2 is a conformal m a p p i n g of W into 1 < [z [ < exp 2zc] [[du[[2 and the images of the components in/~ are circular slits and the images of the components in y radial slits.
The purpose of this paper is to give a complete generalization of the above result to a r b i t r a r y open R i e m a n n surfaces. As a consequence of our work we obtain a new class of conformal mappings of plane regions onto " e x t r e m a l " slit annuli analogous to the situation described above.
We begin with an open R i e m a n n surface W and partition its ideal boundary into four disjoint sets %, ~,/~, y with % a n d ~ n o n - e m p t y a n d ~ , ~ a n d ~r U ~ U ~ closed in the Ker~k- js compactification W of W. Classes of curves ~, :~* analogous to F and F*
(1) This work was supported in part by the National Science Foundation under grants GP 2280 at the University of Minnesota and GP 4106 at the University of California, San Diego.
238 A. M A R D E N A N D B . R O D I I ~
are defined in 1.3. I n Chapter I I we c o n s t r u c t b y means of an exhaustion t h e h a r m o n i c function u corresponding to the p a r t i t i o n ( ~ , zr fl, ~,) a n d in a certain sense d e t e r m i n e the values of u on the ideal b o u n d a r y (roughly speaking, in t h e sense of limits along curves tending to the ideal b o u n d a r y ) . W e also show (II.5) t h e existence of a b o u n d a r y c o m p o n e n t of m a x i m a l " c a p a c i t y " . Corresponding results are derived for a h a r m o n i c function v with a logarithmic singularity at a prescribed point.
I n Chapter I I I we prove o u r m a i n result: a) ~t(:~) = Hdui1-2 a n d b) 2(:~*) = HduH ~. The proof of a) depends on a highly topological c o n t i n u i t y m e t h o d for e x t r e m a l length in which arcs in an e x h a u s t i o n are pieced together to f o r m a n arc in W. This m e t h o d is ascribed to Beurling a n d was developed b y Wolontis [13] a n d Strebel [12]. Using it, Strebel p r o v e d a) in the case fi = 0 . P a r t b) is p r o v e n b y establishing the formula Scdu* = HduH 2 for all curves c E:~* except for a subclass of infinite extremal length. Our m a i n t h e o r e m also yields some uniqueness theorems for u (III.4).
T h e i n f o r m a t i o n we have previously o b t a i n e d is specialized in Chapter I V to the case of plane regions W with ~0 a n d ct each consisting of a single b o u n d a r y contour. W e show t h a t exp 2zt(u + iu*)/HduH ~ is a conformal m a p p i n g of W o n t o a n " e x t r e m a l " slit a n n u l u s contained in 1 < ]z] < R = e x p 2zt/Hdu ][3 such t h a t a) the area of the slits is zero, b ) t h e image of a b o u n d a r y c o m p o n e n t in y is a radial slit (or point), c) t h e image of a c o m p o n e n t in fl which is isolated f r o m y is a circular slit (or point), a n d d) in m a n y other cases t h e image is circular with radial incisions. A n extremal slit annulus is u n i q u e l y characterized (to a rotation) b y t h e following p r o p e r t y : set ~ = ( R log[z[ )-x, t h e n Sc~[dz[ ~>1 a n d SaQidz[ >~
2zt/log R for all c E ~ a n d d E :~*, except for subclasses of infinite extremal length. Our results i m p l y the classical properties of extremal circular slit annuli (y = O) o b t a i n e d b y Reich a n d W a r s c h a w s k i [8, 9] a n d of extremal radial slit annuli (fl = 0 ) o b t a i n e d b y Strebel [12] a n d Reich [7]. E v e n in these classical cases however, the uniqueness p r o p e r t y a b o v e is new.
Contents
I. Preliminaries . . . 239
1.1. Arcs and 1-chains . . . 239
1.2. Extremal length . . . 240
1.3. The classes ~ and ~*: statement of the problem . . . 241
II. Canonical harmonic functions . . . 242
iI.1. Construction of u(z; ~o, ~, fl, ~) . . . 242
II.2. Dependence of u(z; ~o, ~, fl, ~) on ~0 . . . 244
II.3. Boundary behavior of u(z; ~0, ~, fl, ~') . . . 245
II.4. The function v(z; p, ~, fl, ~,) . . . 247
II.5. An extremal problem . . . 249
I I I . The extremal lenght problems . . . 250
I I I . 1.
I I I . 2 . II1.3.
111.4.
IV.
IV.1.
IV.2.
IV.3.
V.
Compact bordered surfaces . . . : . . . 250
Continuity lemma . . . 252
Solution of the extremal length problems . . . 256
Some consequences (uniqueness theorems) . . . 258
Plane regions . . . 260
Definition and extremal properties of the slit mappings . . . 260
The geometry of an extremal slit annulus . . . 263
Extremal properties . . . 267
A dual problem . . . 268 L Preliminaries
I . t . A r c s a n d i - c h a i n s . Given a n open R i e m a n n surface W, its Ker~kjs
compactification, in which each ideal b o u n d a r y c o m p o n e n t becomes a point, will be denoted b y l~ (see Ahlfors-Sario [2]). T h e following topological model of l~ r given b y I.
R i c h a r d s [10] is less well k n o w n b u t c o n c e p t u a l l y quite useful for w h a t follows. T a k e the e x t e n d e d c o m p l e x plane a n d r e m o v e a closed subset S of t h e C a n t o r set f r o m the real axis (a topological model of t h e ideal b o u n d a r y ) . T h e n r e m o v e a countable or finite n u m b e r of disks f r o m the open u p p e r half plane so t h a t t h e y a c c u m u l a t e only to points of S (to ideal b o u n d a r y c o m p o n e n t s of n o n p l a n a r character). N e x t r e m o v e s y m m e t r i c a l l y placed disks f r o m t h e lower half plane a n d identify s y m m e t r i c circumferences b y the correspondence z - ~ L W h e n this c o n s t r u c t i o n has been suitably carried o u t t h e resulting surface will be a topological model of W a n d t h e union of W a n d S will be a topological model of W.
T h e definition of arc a n d open arc in a topological space is standard. W e will use t h e same t e r m i n o l o g y a n d n o t a t i o n for a n arc a n d t h e p o i n t set d e t e r m i n e d b y it.
A relative 1-chain v on W is a countable formal s u m
"If ~ ~ CtT~,
where each c~ is a positive or negative integer, each v~ is a n arc or open are in W, a n d given a n y c o m p a c t set K, T~ N K is n o n - e m p t y for o n l y a finite n u m b e r of i.
The restriction of a n arc or closed curve ~ in l~ r to W will be d e n o t e d b y v fl W. W e see t h a t T N W is t h e n a relative 1-chain on W.
L e t e o = a d x + b d y be a differential on W. W e m a k e t h e following definitions.
(i). Suppose T: ( 0 , 1 ) ~ W is an open arc on W a n d {[t,,tn]} is a nested sequence of in- tervals in (0,1) which a p p r o a c h (0,1). D e n o t e the restriction of T to [tn,t'~] b y Tn. T h e n define
w h e n each t e r m on the right exists a n d t h e limit exists i n d e p e n d e n t of t h e particular ex- h a u s t i o n of (0,1) used.
240 A . M A R D E N A N D B . R O D I N
(ii). I f ~ =
~,c~vt
is a relative 1-chain defineco= 2c, y,,
when all germs on t h e r i g h t exist a n d t h e convergence is absolute.
I f is a linear d e n s i t y a n d v a n o p e n arc I ,
~]dz
I is defined as in (i) a n d always exists ~< ~ . I fz=~c~v~
is a relative 1-chain we defineI n p a r t i c u l a r leo [ = (1 a 12 + [b I~)J Idol ~ a ~ e a r density. I f ~ is a relative 1-chain a n d
Lleol< then Leo e=sts and ILeol <Lleol-
Suppose a triangulation of W is given a n d ~ c ~f is a closed curve or arc with b o t h end points ideal b o u n d a r y points. B y the m e t h o d of simplieial a p p r o x i m a t i o n the relative 1-chain ~ = ~ O W is homologous (singularly) to a simplieial 1-chain vs. I f a is a closed dif- ferential with c o m p a c t support,
W e will also m a k e use of t h e f a c t t h a t if {~l,} is a n e x h a u s t i o n of W (i.e. ~ , < ~ + t a n d
~ is smooth) a triangulation of W m a y be chosen so t h a t ~ for each n appears as a sim- plicial cycle [2]. W e shall use these r e m a r k s later to simplify t h e e v a l u a t i o n of certain integrals.
1.2. E x t r e m a l l e n g t h . I f C is a f a m i l y of rectifiable relative 1-chains a n d
o[dz[
is a Borel measurable linear d e n s i t y defineL(o,c)= foold l,
A(~) = f f w~dx dy, L~(~, C).
~(c) = s u p A ( ~ ) ' 2(0) is called the e x t r e m a l l e n g t h of C.
Traditionally, the elements of C are called " c u r v e s " . I f C is a f a m i l y of arcs on W, t h e same definitions a p p l y except c is to be replaced b y the 1-chain c N W.
L e t C be a family of curves. Following M. O h t s u k a [6] (see also [3]) we s a y t h a t a s t a t e m e n t is true for
almost all curves
in C if the s u b f a m i l yC"
of C for which t h e s t a t e m e n t is false has ~(C') = c~.The following lemma will find frequent use in this paper. In a somewhat different form it is due to Fuglede [3]. From the point of view of Riemann surfaces it also has ap- plications to the theory of square integrable differentials [5].
L v , ~ I. 2.1. Let {~, [ dz
l)
be a / a m i l y o/linear densities on W which satis/y lim A ( e , ) = 0 and suppose C is a [amily o/ curves. Then there is a subsequence {n} such that/or almost all cEC,lira fo =la.I =0.
n-)r
Proo/. Pick the subsequence {n} so t h a t (without changing n o t a t i o n ) A ( ~ . ) < 2 - 3 L Set
B,~ = A,~ U A,,+I U An+2 U ....
E = N B , .
n=l
Then for a n y n, using ~n ]dz] as a competing density for 2(An),
~(W)-l<~(Bn)-l< ~
/~(At)-1~--~ ~ 2 - t = 2 1 - n - - - ~ 0i=rt t =rt
(here we are making use of the well-known result that if a class of curves r is contained in a countable union of classes [.J 1~= then 2(I') -1 ~< ~2(I'n)-l; see [6]). Hence 2(E) = c~. If for some c E C, lira sup 5c Qndz > 0 then c belongs to all B~ and therefore to E.
1.3. T h e c l a s s e s 3 a n d 3*: s t a t e m e n t o f t h e p r o b l e m . Let W be an arbitrary open Riemann surface and partition the ideal boundary into four disjoint sets ~o, ~, fl, Y such that
(i) ~0 and ~ are non-empty
(ii) ~o, ~, and ~0 U ~ U fl are closed sets in the compaetification W of W.
Define the classes 5, 3" (or ~(~0, ~, fl, r), ~*(~o, ~, fl, 7)) as follows.
:7= (~: v is an arc in W - 7 with initial point in %, end point in ct}
:~* = {z: v is a countable union of closed curves in W - ~ o - ~ - f l such t h a t (a) all limit points of v are contained in 7, and
(b) no component of ( l ~ r - 7 ) - v contains points in both ~ and ~}.
Given 9 E:7* let W1 be the union of the regions in (~V- 7 ) - v which contain points in
242 A . M A R D E N A N D B. R O D I N
A
u0 and let TI be the relative boundary of W1 so t h a t ~1 is contained in ~. Then TI satisfies the following property
(c) If ~t is a closed curve in T1, then ~1 - T t does not satisfy (b).
The main problem of this paper is to find ~(~), ~(~*). Therefore it is no restriction to assume the curves in ~* also satisfy (c). Then the curves in ~* m a y be oriented so t h a t ~0 lies to the left.
The method of solution of these extremal length problems is really dictated b y the assumption (ii) on (~0, ~, 8, 7')- I n Chapter V we will briefly consider the problem of finding /~(~) and ~(~*) when (ii) is replaced b y
(ii)' ~0, ~, and ao U ~ ~ 7' are closed in W.
The method of finding the solution is different, and even when the second partition is obtained from the first b y adding to 7' and subtracting from 8 the minimum number of points to make (ii)' true, the solution m a y be different.
II. Canonical Harmonic Functions
I I . t . C o n s t r u c t i o n of u(z; ~0, ~, 8, 7')- We begin with an arbitrary Riemann surface W and a partition (~0, u, 8, 7') of the ideal boundary of W as described in 1.3 above. Assumption (ii) implies t h a t there is a J o r d a n curve in W separating ~0 and ~. In this section we will construct the harmonic function u(z) = u(z; ~o, ~, 8, 7') determined b y the " b o u n d a r y " condi- tions u - 0 on a0, u = l on u, u = constant on each component 8i in 8 with SB~ du* = 0 and
~u/an = 0 along 7'. These conditions are to be understood as limits in a sense to be made precise below. Finally we wish to emphasize t h a t convergence (of functions and differentials) is considered only in the Dirichlct norm.
Let {~2n} be an exhaustion of W such t h a t for each n, ~ separates ~0 from z and each component of ~ n is a dividing cycle not homologous to zero. Suppose ~ is a component of some ~ ; then q is the relative boundary of a subregion S of W - ~ . If r~ is an ideal boundary component of S we shall call ~ a derivation of a.
Using the following rules divide 0 ~ , into disjoint collections of components ~n, ~ , 8n and 7'n.
A - l ) a~, consists of those components of O~n which have a point of ~0 as derivation.
A-2) :on consists of those components which have a point of :~ as derivation.
A-3) A contour of ~ n belongs to 7'n if and only if the only ideal boundary points which are derivations of it lie in 7'.
A-4) 8in consists of the remaining contours of ~ .
L e t ~ n denote the possibly noncompact region obtained b y adjoining to ~= the non- compact components of W - ~ which are bounded b y curves in y.. We orient the b o u n d a r y contours so t h a t ~ = ~= + fin - ~0~. Choose an exhaustion {~n~ } ~r 1 of ~ n SO t h a t ~ , ~ = :r +fin + Y ~ i - ~0n- Thus for all n, y ~ is homologous to yn.
Define the harmonic function u.~ in ~ b y the b o u n d a r y conditions B - l ) un~=O on ~0n, uni = 1 on gn
B-2) i~unJ3n =0 along Yn~
B-3) u~t is constant on each component flnj in fin with the constant chosen so t h a t S p.~du~t =0. $
We will first show that limt-~:cun~=un exists in ~n and is independent o/ the exhaus.
tion {~n~} o / ~ .
:For j > i the equation
dun,)o., = J0~ un, u:, = fo = II d n, I1 .,
implies t h a t
]ldun,-dun,]15.,<
Hd~n,I]5.rlld~n, ll5.. (1)
I n
particular,
Ild/n,]lh., is increasing with i.Let v be the harmonic function in ~no = ~ . defined b y the boundary conditions v = 0 on ~0., v = 1 on ~ n - ~on. Then since
we find t h a t
]ldu~,-dv[l~~ ~< Ildv]lh. - Ildun,llh.,,
a n d therefore Ildun,]l is uniformly bounded. I t follows from equation (1) t h a t Iim,_+r162 un, =u.
exists. F u r t h e r m o r e lim,..+~clldun,l[~,,=Hdunl[~,,. To prove this last assertion set A = lim,_~:r ]ldun,lJ~,,. Then using a simplified notation, let i - + ~ , i >?', in the inequality
I I ~ , - d~,ll~ < Ild~,ll, ~ -IId~.,ll~
to o b t a i n
Ildu,~ -dun,ll~ <-< A -Ildu~,ll$.
Our assertion now follows upon letting 7"-+ ~o in the inequality:
I IId~-d~n, ll,-Ildu~ll,I < Ildu~,ll, < IId~-dun,ll,
+IId~:ll,
Clearly u n is independent of the particular exhaustion {~.i} of s and un is 0 on %n, 1 on ~. and constant on each component fln~ of fin with .f~,~du* = 0 .
244 A . M A R D E N A N D B . R O D I N
N e x t we show that lim dun =oJ exists as a n exact harmonic di//erential independently o/
the exhaustion {~,'}. I f n > m then s
~(~,', n ~m) = am +fin--~'.,, +7,,, n ~,~,
and tim, ~m, a~ are homologous to components in fl,', Y,'*; a ~ , fl,', 7,'5 a~, fl,', Y,'~, respectively.
Hence since lim,_~]ldu,'-dundl~,, , = 0 , ( d u , ,
Therefore
II ~.,-
~unllL <II
du.IlL- II d~,'llk
(2)and we see t h a t Ildu~llL is decreasing as m increases and limm_~ dum exists. The harmonic limit differential is exact a n d we denote it b y du, u being unique only up to an additive constant. We also see t h a t l i m l l d u m l l ~ = Ildull. The question arises, does lim u , ' = u exist for suitable u? This is false if
Ildull
= 0 b u t t ~ o u t to be c o = c o t ifI1~211
> 0 . T h e proof follows easily from later results (Theorems II.3.2 and III.3.1). F o r the present wedo not need this fact.
I f { ~ } is another exhaustion of W, then given k there exist m and n such t h a t ~ <
~ m < ~ . I t follows t h a t ~ < ~ m < ~ a n d we see t h a t indeed du is independent of our choice of exhaustion (~n}.
We will have occasion to use the following simple observation. F o r each n we can find an integer in such t h a t
Ildu,'ll~-II~u~,.llo.~ < - . ~b 1
Thus on setting v," = u , , , a n d replacing ~,',, b y ~," we see t h a t
limlldvdl~.
= Ildull.By
pas- sage to a subsequence we m a y assume the ~n are nested and thus f o r m an exhaustion.II.2. D e p e n d e n c e of u(z; ~o, ~, f , Y) o n :t o. For m a n y applications ~0 is a finite collection of m u t u a l l y disjoint, piecewise anlytic J o r d a n curves imbedded in a R i e m a n n surface W in such a w a y t h a t ~0 does not separate W into two non-compact components. I f this is the case the exhaustion {~n} m a y be t a k e n to be an exhaustion of W - ~ 0 less compact regions bounded b y ~ (if any) so t h a t a ~ , = an eft," + Y n - a~, for all n (~0 used in this con- nection m a y actually consist of b o t h sides of the curves a0). Then u =u(z; ~ , a, fl, Y) m a y be constructed as above, and is harmonic in W - compact regions bounded b y ~0.
Now suppose t h a t we are given another such collection of curves ~ disjoint from ~ . Choose a corresponding exhaustion { ~ } so t h a t ~ = ~," +fin +7," - ~ and construct the harmonic function u' =u(z; ~ , a, fl, 7). As a consequence of the observation made a t the
end of I I . 1 , we m a y assume t h a t {~n} a n d { n) are chosen so t h a t
limlldvnll
=Ildull, limlldv~ll
= Ildu'll(see
the n o t a t i o n used there) a n d ~ o c ~ , ~ c ~ , for all n.Setting K , = ~ N ~ we see t h a t ~K~ = ~ + fl, + y~ - ~0 - ~ a n d
v n d v n =
dUn*- v.dv':
= Ildv'. IIL,.-(dv=, dvn)K. = "* "*
K n ~ o ,J 0~o
Hence
IIdv=- dvn
I1~ < Ildvnll~- IIdv'n I1~ + 2 vndvT.
I] ~a
We conclude t h a t if lim II
dvn
I1o~= II du II = o, t h e n lim v, = 0 a n d since d r 7 >1 O, ~ , d v 7 = IIdv'~
II ~,lim II
dvn- dv'~
I1~ = - lira IIdv'~
II ~ = - II du' II ~ThereforeHdu'll
=o,
a n d we h a v e p r o v e d m o s t of t h e following theorem.THV.OREM II.2.1. Given ~o and ~ as above, but not necessarily disjoint, then u(z; ~o, ~, fl, Y) = 0 i/ and only i~ u(z; ao, o~, fl, y) = 0 .
Proo/. W e h a v e just p r o v e d t h a t if g0 a n d ~ are disjoint the conclusion holds. I f ao a n d ao are n o t disjoint, choose ao' disjoint f r o m b o t h =0 a n d ~ a n d c o m p a r e u a n d u ' with u(z; ~4', ~, ~, ~').
I I . 3 . T h e b o u J a d a r y b e h a v i o r of u(z; ac0, co, fl, y). I n this section we show t h a t in a certain sense u(z; :co, ~c, fl, y) takes t h e expected values on t h e ideal b o u n d a r y .
LEMMA II.3.1. Notation as in II.1. Suppose c is a union o/analytic Jordan curves divid- ing ~,~ into two regions A z and A 2 in such a way that OA z consists only o/c and elements o/
ft, and~or Yn" Then/or all ZEAl,
0 < minum(p) ~< uni(z) <. m a x u m ( p ) < 1.
p~C ~EC
Proo/. To simplify n o t a t i o n we will omit t h e subscripts a n d superscripts on u a n d ~ . W e prove the right inequality; the left one can be t r e a t e d similarly. I f our assertion is false t h e n maxzeAlu(z) is a s s u m e d on a c o m p o n e n t ~ of ~ which belongs to fin or Yn. Pick a point p E I n t A 1 such t h a t u(p) = k > maxz ec u(z) a n d let 1 be t h a t p o r t i o n of the level curve u = k (not necessarily connected) t h a t lies in A 1. N o t e t h a t 1 does n o t m e e t c. W e m a y as- sume t h a t 1 does n o t pass t h r o u g h a critical p o i n t of u a n d hence does n o t meet a n y com- p o n e n t of ~A 1 in fin. H o w e v e r l m a y meet c o m p o n e n t s of ~A 1 which are in Yn. L e t A~ be the c o m p o n e n t of A 1 - l which contains ~ in its b o u n d a r y . Since u ( z ) > k for z E A l , du* < 0 along l' = l N ~A~ (when l' is oriented so t h a t A1 lies to the left) a n d we m u s t h a v e j'v du* < O.
On the other hand, l' possibly together with pieces of c o m p o n e n t s of aA~ in Yn is homologous
16 - 6 6 2 9 4 5 A c t a m a t h e m a t i c a . 115, I m p r i m 6 le 11 m a r s 1966.
2 4 6 A. MARDEN AND B. RODIN
to a combination of components of ~A~ which are in ~n and Yn. Because du* =0 along components in Yn, we must have then Sv du* =0. This contradiction proves the lcmma.
THEOREM I I . 3.2. I] 2.(:~)<~ there exists a function u=u(z; %, ~ , f l , $ ) s u c h that a) lim un=u (see 11.1) and ]or almost all curves TE:~
f d u = l .
b) For each component fl~ Eft, there ezists a number b, 0 <b < l, such that/or almost all arcs T in ~ V - ~ with initial point v(0) in W and end point on fli,
f du =.b - u(v(0)).
(This result holds/or ~o and o~ with b replaced by 0 or 1 respectively.)
Proof. The class of curves to be considered in b) m a y be written as the union of classes Fn such that for v EFn, v is an arc in W - ~ with v(0) E ~n- Hence it is sufficient to prove b) for curves v with 3(0) E ~ r Let c, be the component of 0~n determined b y fl~ (or % or ~) - - s e e the notation in II.1. B y eliminating a class of curves of infite extremal length we m a y assume that there are only a finite number of components of 3 N ~ n which meet more than one component of ~ n , for all n. If 3 leaves ~ n by crossing a component c ~= % of 0~n then the next return of 3 to ~ n is b y crossing c again; 3 leaves ~ n for the last time b y crossing cn.
Let u n have the value bn on cn. I f c n is determined b y % or ~, then bn is 0 or 1 respectively.
The differential du is uniquely determined as lim dun. Set un = 0 in W - ~ n . The cor- responding linear densities en I dz ] = [ d u - dun ] ( -- [grad (u - un) I I dz I) satisfy
A(on) = [[du-dun[[ 2 ~ 0 as u->oo.
Lemma 1.2.1 asserts t h a t for almost all 3, there exists a sequence {m}, not depending on 3, such t h a t
l i m f d u - ( b ~ - u m ( 3 ( O ) )
=lim[.fdu-dum]
~<lim f~ [du-dum] =0.(This formula contains the assertion t h a t S~ du exists. A simplicial approximation to 3 m a y be used to evaluate ~dum.)
Now suppose that 2(:~)< oo. B y assumption there is a J o r d a n curve J separating ~0 from ~. Since 2(:~)< co the extremal length of the class of curves in t ~ - ~ which travel from J to % is finite. Given a point p E W we can connect each such curve (i.e. arc) to p b y an arc from p to J . Since lim j'r [ du - dun [ = 0 on every compact curve c we can reapply the reasoning above to obtain
This proves t h a t limm->~ u,n(p) exists and more generally that, for the original sequence, lira ~n(P) exists. Hence the function u is uniquely determined as lim Un.
N e x t we obtain from above
lim
~ du-(b,,-u(3(O)))l=O.
Hence lim bm and more generally b = l i m b, exists. Since a 0 and a are closed sets in ~V, every
fl~Efl
is isolated from a 0 and a. L e m m a II.3.1 and the m a x i m u m principle imply therefore t h a t 0 < b < 1.The remainder of b) and a) now follow easily using the above methods.
Remark.
I n the case t h a t 3 is in I ~ - fl,f du=
lim U(Z) - lim U(Z),z-~v(1) z-=~v(0)
where 3(1) and 3(0) are the end point and initial point respectively of 3 in I~.
II.2.~.. T h e f u n c t i o n
v(z; p, a, fl, ~).
Suppose now t h a t we have a partition of the ideal b o u n d a r y into three disjoint sets a, fl, ~ such t h a t a and a U fl are closed in the eompacti- fication ~g of W and a is not empty. We will construct a harmonic functionv(z)=
v(z;p, ~,B,T)
with a singularity log]z] a t a prescribed pointp (z(p)=0)
a n d behavior on~, fl, y the same as t h a t of u.
As in II.1 we start with an exhaustion (s so t h a t p e f ~ , for all n and divide 0~n into three sets an,
fin, 7n
b y the rules A-2), A-3), A-4). Then we construct ~ n with ~ n =~n +fin and take an exhaustion (~n~} of ~ n so t h a t ~ n i = gn +fin +?n~ where~n~ is homo- logous to 7n. Let vn~ be harmonic in ~n~ except for the singularity log]z] at p and satis- fying the b o u n d a r y conditions
Vn~
= 1 on an, vn~ = c o n s t a n t on each component fit in fin so t h a t~dv*~ =0,
andavnJ~n=O
along ?hi.First we show t h a t lim~o~ vn~ =vn exists. E x t e n d vnl to ~ n b y setting vnl = 0 in s g2~1. Dropping the subscript n, we find for
j>i,
(g(v, - v,), d(v, - v,) )~, = f , (v~ - v,)d(Vl - V,)* + f , vjdv: = f , v, dv~ = - l[ d(v, - v,) [[L,.
Hence ]]
d(vj - v,) ]]~,
= ]]d(vj - Vl) + d(v~ - v~) ]]~,
~< ]]d(vj- v~) ]]~j
- ]]d(v, - v~) ]]~.
Thus
]]d(v,-v~)]]a~
is monotonically increasing in i; it is also uniformly bounded. F o r let w have the singularity log] z] at p and b o u n d a r y value 1 on ~g21. We find t h a t248 A . MARDEI~" A N D B . R O D I N
I I ~(v~- w) ll~, = II ~(w- v,)I1~,-lid(v,- vl)II~,.
H e n c e II d ( v , - vl)II~, is u n i f o r m l y b o u n d e d a n d lim,_~ v., = v. exists in ~ n with lim~-~clid(v. - vm)lira = 0 (u., = 1 on a . for all i).
N e x t we show t h a t lim v n =v exists in W as a harmonic /unction with singularity log ]z ] at p. L e t A be the p a r a m e t r i c disk I z ] ~< 1 center at p a n d define v 0 as
v 0 ~ l + l o g l z ] , z E A , v 0 = 0 , z ~ A . On breaking the Dirichlet integral into t w o p a r t s we find for n > m,
( d ( v . - v o ) , d ( v m - v0))a~ = lira ( d ( v . ~ - v o ) , d ( v m - v0))n., ~
= - foa ( v m - 1)d.~ = II
d(,m-%)ilk.
Uence II a(v.- v~)Ilk: < II d(,.-,o) llk-II ~('~-'0)Ilk..
Since Hd(v.-vo)l[~. = - ~ o ~ ( v n - 1 ) d v ~ , we see t h a t either l i m v n = - c ~ or l i m v . = v exists as a h a r m o n i c function in W - {p} with singularity log I z ] a t p.
As in I I . 1 , v is i n d e p e n d e n t of the e x h a u s t i o n {~n}.
Suppose v~--v(z; p, ~, fl, y) a n d v z have the expansions at p
v=loglz I + c + o 0 ) , v~=log[z I +e~+o0).
Then ck > ek+l/or all k and lira ck = c.
F o r set v o = l + l o g ( [ z l / r ) in the disk A : l z I ~<r. W e find
= - 2~ log r - 2gck - 2~ + o(1).
Since Hd(vk--Vo)HUk is increasing, c~ is decreasing. Since lim vk=v, lim ck=c.
Choose k so small t h a t the level curve ~0 on which v = k b o u n d s a relatively c o m p a c t subregion K of W containing p. Then u(z; ao, a, fl, 7) = ( 1 - k ) -1 [v(z; p, a, fl, 7) - k ] in W' =
W - K .
F o r we m a y take the e x h a u s t i o n { ~ . } of W so t h a t K ~ ~n for all n. Use t h e n o t a t i o n Un~ for the a p p r o x i m a t i o n s to u(z; ao, a, fl, 7) with respect to the exhustion { ~ = ~ . - K } of W' a n d v~t for the a p p r o x i m a t i o n s to v(z; p, o~, fl, 7) with respect to { ~ . } as c o n s t r u c t e d in t h e p a r a g r a p h s above. W e find
I d [ u n , _ l l k ( v n _ k ) ] l : , , - 1 k f ~ , ( v m _ k ) d ( u n ' 1 "
Letting i ~ ~ a n d t h e n n - + ~ we obtain
d [u(z; ~o, o:~ fl, 7) - l--l k (v(z; p, ac, fl, 7) - k)] ~ . = 0 since convergence is uniform on ~o.
Using the results of I I . 2 we see t h a t v(z; p, ~, fl, 7) = - ~' i] and only i] v(z; q, (u, ~, ~) = - oo/or any pair o/points p and q.
The b o u n d a r y behavior of v(z; p, ar fl, 7) is determined exactly as in II.3.
I I . 5. A n e x t r e m a l p r o l } l e m . Partition the ideal b o u n d a r y into three disjoint sets ~o,
~' and 7' where/~' is closed in ~r and fl' U 7' is isolated from ~0- Suppose ~ consists of a single b o u n d a r y component and allow :r to range over the set S =fl' U 7'. When :r is chosen, set f l = f l ' - ~ , 7 = 7 ' or f l = f l ' , 7 = 7 ' - - ~ , depending on whether ~Efl' or ~ET'. Indicate the dependence of u(z; zoo, ~, fi, 7) on zr b y the notation u~. Thus ~ ~
lld ll is
a real valued function on the compact subset S of It 3.THEOREM IX.5.1. The /unction ~ [ I d u = l l r sc. Hence there i~ a component o~ES which azim z ll u ll.
Proo/. L e t a = l i m s u p a . ~ [ I d u a , I I. We take a sequence :ca such t h a t limnlldu~ll = a, an---> ~.
Consider u~ and let ~ k be one of the regions in the definition of u~ with u~ the ap- proximation to u~ in s For large n, uk is also an approximation to u ~ ; from I I . 1 we have, for sufficiently large n,
Ildu ll. > lldu ll,
and thus, II I1- >/a.
This implies t h a t I] du~ II >/a.
There is a corresponding theorem for the functions v(z; p, a, fl, 7) as follows. L e t fl' and 7' be a partition of the ideal b o u n d a r y into two disjoint sets so t h a t fl' is dosed in W.
L e t a be a single component ranging over the ideal b o u n d a r y and once a is chosen, set f l = f l ' - a , 7 =7' or fl=fl', 7 = 7 ' - a , whichever is relevant. Writing % for v(z; p, ~, fl, 7), a b o u t p, % has the expansion
% = l o g l z I +c~ +o(1).
250 A . M A R D E N A N D B . R O D I N
THEOREM II.5.2. The/unction ~-~ca is u.s.c. Hence there is an ideal boundary point o: which maximizes %.
Proo/. Set b = l i m sup~,,_~ ca, and choose a sequence :on such that ~-->~ and lim~ c~ -- b.
Let {$~k} be an exhaustion used to define va. Let v k be the approximation to v~ in g~-~.
B u t for large n, vk is also an approximation to v~. Hence ck ~> c~ for sufficiently large n.
I t follows t h a t ck ~ b and then c~ ~> b.
Note that c~ depends on the local parameter z. I f w = w(z) is another local parameter with w --0 at p and c~ corresponds to c~ for w, then
Thus the component which maximizes % does not depend on z.
I H . The Extremal Length Problems
I I I . i . C o m p a c t b o r d e r e d s u r f a c e s . Suppose the boundary components of a compact bordered Riemann surface ~ are divided into four disjoint sets ~0, ~, 8, 7. Let u = u(z; g0, ~, 8, 7) be the harmonic function in ~ determined b y the boundary conditions u = 0 on ~0, u = 1 on ~, u = c o n s t a n t on each component fl~ in fl in such a way t h a t S~,du* = 0 , and
au/~n = 0 along 7.
Denote the compactification of ~ b y ~ (then ~0, ~, 8, Y are each interpreted as a finite point set in ~ ) and define the classes F, F* of curves as in 1.3. I n the present situation however, when c E F or F*, we m a y assume c fl ~ has a finite number of components.
Denoting the extremal length of F and F* b y ~t(F), 2(F*), respectively, the object of this section is to prove the following Lemma.
L E g a A I I I . l . 1 . a) 2 ( F ) =
Ildull%
b) 2 ( F * ) =Ildull ~.
P r o o / o / a ) . The proof is accomplished b y finding a parametrization ls of level curves of du* such t h a t ls E F for almost all s.
The boundary ~ is oriented so that ~ lies on the left. Fix a point p on ~0 as the origin and fix other points on ~0 so t h a t there are two points specified on each component of ~ . Then using these points determine a route which traverses a o exactly once in the positive direction, beginning and ending at p. Since du* is strictly negative along this route, setting u*(p) = 0, define u* (z), z E g0, so that as z passes along this circuit u*(z) takes on each value s, o>~s~> -[[dull ~, exactly once.
W e will always orient the level curves of du* so t h a t u is increasing in the positive direction. F o r each 8, -IIdull ~ <8 ~<0, a single level c u r v e / 8 of u* leaves ~0- I n the succeed- ing p a r a g r a p h s we shah f o r m u l a t e rules for dividing the i n t e r v a l - I l d u l l 2 <8 <.0 into t w o c o m p l e m e n t a r y sets, S a n d C.
I f the connected level curve I s passes t h r o u g h a critical p o i n t of u in ~ ( = closure of
~ ) , p u t s in C.
I f 8 ~ C, t h e n ! s ends a t either ~ or ft. I f 18 ends a t ~, p u t s in S a n d write ! 8 s i m p l y as ls.
Suppose ! s ends a t some c o m p o n e n t fit Eft. I f P8 denotes the end p o i n t of 18, follow fit f r o m Ps in the direction of increasing du*. W h e n fit is t r a c e d in this m a n n e r the arc 18 a n d hence ~ lies to the left. L e t qs be the first p o i n t for which Sdu* = 0 , the integral being t a k e n over the arc on fit f r o m P8 to q~.
I f q8 is a critical p o i n t of u, p u t s in C. Otherwise we see t h a t a level curve l; of du*
a t qs begins a t q8 since if u=c~ on fit, the fact t h a t u is <c~ to the left of P8 implies t h a t u i s > c t to the left of qs. W e will refer to l~ as the c o n t i n u a t i o n of 18.
I f l~ goes t h r o u g h a critical p o i n t of u on ~ , we p u t 8 in C. Otherwise we r e p e a t t h e a b o v e procedure. N o t e t h a t l; c a n n o t r e t u r n to fit w i t h o u t passing t h r o u g h a critical p o i n t of u.
F i n a l l y we end with the
following
situation. E a c h s, -IlduH 2 < s 4 0 , is either in C or there is a n ls, which is a finite union of connected level curves of du*, one the c o n t i n u a t i o n of the preceding, which runs f r o m ~0 to a. F o r 8 6 S, each 18 can be r e g a r d e d as a n arc inr u n n i n g f r o m e0 to e, t h a t is 18 E F .
W e m a k e the following two observations.
(1) I f 81, 82ES , s1~82, t h e n ls, n 18~=0. T o p r o v e this it is enough to observe t h a t if 18, a n d 18~ end on the s a m e c o m p o n e n t fit Eft, the c o n t i n u a t i o n of 18, m u s t be different f r o m t h e c o n t i n u a t i o n of 18~. Indeed, if the direction on fit d e t e r m i n e d b y 18, is the s a m e as t h a t d e t e r m i n e d b y 18,, t h e n qs, =qs, would require qs,=Ps,, which is impossible. I f the direction on fit d e t e r m i n e d b y 181 were opposite to t h a t d e t e r m i n e d b y 18, y e t q81 = qs,, t h e n ~ would lie b o t h to the left of qs, a n d to the right of qs,, a n a b s u r d i t y .
(2) C consists of a finite n u m b e r of points. F o r u has only a finite n u m b e r of critical points in ~ a n d only a finite n u m b e r of J o r d a n arcs along which du* = 0 pass t h r o u g h each critical point. I n other words, e x c e p t for a finite n u m b e r of 8, 18 can b e continued until it reaches ~.
E x c e p t a t the finite n u m b e r of critical p o i n t s of u, u + i u* can be used as a local p a r a - m e t e r a n d ~ can be p a v e d w i t h little rectangles d e t e r m i n e d b y the level curves of u a n d u*. I f ~ I dz ] is a linear density, f r o m t h e Schwarz inequality if s e S,
252 A . M A R D E N A N D B . R O D I N
ft QIdzr= ft ~iu~< (f~ ~2du fz du) = f~ ~ ~du"
I n t e g r a t i n g in s from - H
du
I[' to 0, s E S, a n d we obtaina n d hence 2(F) ~< ][du
H -~"
However, using~[dz I
= ]grad u []dz[
as an admissible density we obtainthus obtaining the proof of a).
Proo/o/b).
The level locus (not necessarily connected) u = c for O < c < l contains a curve in F*. The proof is completed b y a repetion of the a r g u m e n t using the Schwarz inequality given immediately above.I I I . 2 . G o n t i n u i t y le~a~aa. We will now m a k e use of an extremal length technique due to Beurling and developed b y Wolontis [14] and, most closely approximating our present context, b y Strebel [12].
L e t {~n} be an exhaustion of W of the t y p e considered in II.1, a n d let _F n be the class of curves in ~n--~n which go from ~ to ~n, via possibly some contours in fin (see I I I . 1 ) . More precisely, l E F n if and only if the domain of I consists of a finite union of closed inter- vals [%, all 0 [az, aa] U ... U [as_i, as] with a 0 < a 1 < . . . < as, I is a continuous m a p p i n g into
~n--~n, l(ao)E~,, l(as)Ean,
a n d for oddi<j, l(a~)
and /(a~+l) belong to the same compo- nent of fin.L~MMA III.2.1. limn_.~ 2(Fn)~>2(~).
Proo]. Restatement.
We shall define a family :~' of relative 1-chains on W such t h a t (a) restw(:~ ) c 5 '(b) lim 2(Fn) ~>2(:~')
(c) ,t(T)=~(:~),
where restw(:~ ) = {1N W :
leJ}.
Once this is done the proof of the l e m m a will be complete. Recall t h a t s was obtained 9 from g2n b y attaching to it all components of W - ~ whose boundaries belong to Yn.
L e t :~n be the family of curves in C1w~ . which go from g0n to gn via possibly some fin's.
More precisely, l E : ~ if a n d only if the domain of l consists of a finite union of closed in-
tervals [a0, al] U [a S, an] U ... U [aj,x, aj] w i t h a 0 < a l < ... < a j , l is a continuous m a p p i n g into t h e closure (with respect to W) of ~ n , l(ao) E no,, l(aj) E an, a n d for all o d d i < j , l(a~) a n d l(ai+l) belong to t h e s a m e c o m p o n e n t of fin. T h e n F~ = : ~ so 2(Fn) >~2(3;=). H e n c e i n s t e a d of (b) it suffices to p r o v e
(b') lim 2(:~n) ~>~(:~').
W e wish to replace (b') b y a n o t h e r condition. Choose a n y x < ~ ( ~ ' ) . Choose a linear
2 t
d e n s i t y ~ldz] such t h a t L ( ~ , ~ ) > x a n d A @ = I . T o p r o v e (b') it is enough t o show s u p . L~(:~, ~)>~x. I f t h a t failed to hold t h e r e would exist a subsequence along which L~(:~,, ~) h a d a limit y < x. L e t L(l, ~) = ~, ~ ]dz [. If we can p r o v e t h a t to each e > o t h e r e is a n l(e) ~ :~' satisfying
L~(l(e), ~) < y + 7 ~ t h e n we would h a v e the desired contradiction since
y < x < L 2 ( ~ ', ~) ~<L~(/(e), ~) ~<y+7~.
(b") G i v e n a d e n s i t y ~ [ dz I on W with L~(:~, ~) ~ y as n ~ ~ . T h e n to each e > o there is a n l(e)E :~' to be defined according to (a), (e) such t h a t
L2(l(e), ~) ~< y -t- 7~. (3)
Some notation and terminology. Given 1NE:~N a n d n ~<N, we wish t o define a sort of restriction of 1N to CIw~n, to be d e n o t e d b y lNII ~ , , w i t h t h e p r o p e r t y t h a t lN[[ s e:~n.
T h e r e is a g r e a t e s t t for which/~(t)Eao~; call it t r L e t t 2 be the smallest t for which lN(t) E ~ n a n d also t > t 1. T h e n ln(t~) is on some c o n t o u r of fin (or possibly an); call the con- t o u r c~. Set t 3 = g r e a t e s t t for which 1N(t)Ec~. W e continue this w a y a n d o b t a i n a n e v e n n u m b e r of stopping times t l < t 2 . . . < t k , a sequence of s t o p p i n g points IN(t1) .. . . . /N(tk), a n d a contour sequence non =cl, c 2 . . . c(~+~)/2 = an of distinct contours on ~ n such t h a t 1N(ti)EcE(j+~)12 J ( j = l . . . k). Define 1NH~ . to be the restriction of IN to [ts, t2] U [ta, t4] U ... U [tk-1, tk].
De/inition o/if'. A 1-chain l' on W will belong to :~' if either l' = l N W for s o m e / E f t , or if l' is a continuous m a p of a n open dense subset of (0,1) into W such t h a t :
F ' - I . I f t o is n o t in t h e d o m a i n d o m l' of l' a n d 0 < t 0 < l , t h e n t h e r e exist sequences
r _ ~ t . _ ) .
{r~ }, {s~} in d o m l' such t h a t rn S to, s~ ~ t o a n d a p o i n t ~e e fl such t h a t l (r~) ~e, 1 (sn) ~e.
I f t o = 0 (resp. 1) we require only a sequence {sn} (rcsp. {rn}) f r o m d o m l' w i t h s n ' ~ 0 (resp.
r , 7 1) a n d l' (s~)-~ a 0 (resp. l' (rn)-~ a).
F ' - 2 . T h e r e is a canonical e x h a u s t i o n {s such t h a t l'll~N e :~N for each N >~ 1.
F ' - 3 . I f t E d o m l' t h e n there exists N such t h a t t E d o m l ' H ~ n for all n>~N.
254 A. M A R D E N A N D B . R O D I N
Proo/o/(a) and (b'). Condition (a) is part of the definition of :~'. To prove (b') suppose given an e > 0 . B y passage to a subsequenee of {l.} we m a y assume t h a t
]L2(:~, O)-Y] <e/2" (n>~l). (4)
Whenever a subsequence of {In} is extracted and the notation is unchanged we tacitly agree t h a t {fl~} shall refer to the corresponding subsequence of {n}.
Choose l~ 6 : ~ such that
[L~(ln, O)-Yl <e/2" (n>~l). (5)
A subsequence of {/~}, after some modification, will be used to construct l(e).
The first step is to find a subsequence {/,~} of {ln} such that all ln, l ~ N ( n , >~V) have the same contour sequence on 8~N. Since there are only a finite number of possible contour sequences o n ~"~1 We m a y select a first subsequenee of {l, }, all elements of which have the same contour sequence on ~ 1 . B y induction we obtain for each N a subsequence of the preceding one, all of whose elements follow a common contour sequence on 8s The dia- gonal process yields a subsequenee with the desired property. We shall not change notation, but denote it still by {/=}. Note t h a t (5) continues to hold.
The next step will be to modify each 1, so t h a t not only will all l~ll~ N (n~>h r) follow the same contour sequence but furthermore, 1 n ]] ~ , - 1 and 1,_l][~n_l will have the same sequence of stopping points on as r To do this we use the diagonal process to find a preliminary subsequence, again called {/~}, with the following property: Suppose lN has k stopping points on O~N. Then for each i ~<k the ith stopping point P , of/.]]~N (n >~h r) gives rise to a conver- gent sequence of points {P,} on a contour of 8~N. Now, we put a topological disk around the limit point of this sequence, the circumference of which has very small 0-length. The actual length will be determined below. Note, however, t h a t it can be required to be ar- bitrarily small. Indeed, the extremal length of all J o r d a n arcs in a punctured disk which surround a fixed point is zero, and hence for a n y o ldz] there is such an are of arbitrarily small 0-length.
For each h r we have as m a n y disks on 3~N as there are stopping points for/,]]~N (any n ~>hr). Choose their circumferences so small that the total 0-length of them is <e/2 ~.
B y the diagonal process we can achieve a situation were each stopping point of ln]]~N on 0~N is inside its appropriate disk for all n, h r with n/> h r. For each disk pick a point on the intersection of its circumference and the corresponding contour; call such a point a dis- tinguished stopping point. B y a modification of 1. we mean the result of replacing part of its path inside a disk by a path on the circumference of the disk. Now modify 11 so t h a t all its stopping points are distinguished; in general, modify In so t h a t the stopping points of /~[[~-~-t on 0~,,_~ and / , [ [ ~ on O ~ are distinguished. Denote the modified sequence
again b y {l.}. T h e q-length of l,, has been increased b y no m o r e t h a n el2 =-1 + e l 2 = as a result of modification. F o r the p r e s e n t sequence {/~} e q u a t i o n (5) m u s t be changed t o
IL(Z=, e)-Yl <4e/2-. (53
These modifications can be accomplished so t h a t each n e w In r e m a i n s in : ~ .
B y induction, for each n r e p a r a m e t r i z e l~ so t h a t d o m [~]]~=_ 1 = d o m ln_ r T h e n doml~_ 1 consists of a finite n u m b e r of closed intervals It I, t~] U [t3, t4] U ... U [tk-1, t~] a n d l=_l(t~) = l~(t~) (1 <i<.k).
W e are n o w r e a d y to construct l(e). On dora 11 set l(e) = l 1. I n general, if l(e) has b e e n defined on d o m ln-x set l(e)=l~ on d o m / , - d o m l~_ 1. T h e n l(e) is a continuous 1-chain on W. I t s d o m a i n is a n open subset of (0,1) which, b y r e p a r a m e t r i z a t i o n , we m a y a s s u m e to be dense.
T o e s t i m a t e t h e e-length o f l(e) n o t e t h a t t h e e-length of l, restricted to d o m l ~ - dora l=_ 1 is < (6e/2n) 89 Indeed,
L2(l~, e) <Y +4~/2~
b y (5') and, s i n c e / ~ [ d o m ln_~=l~l[~_~Ey~_~,
L~(l=ll~2=_~,
e) >~L2(:~,_1, Q) > y - e / 2 ~ b y (4). H e n c e L2(l(e), ~) <L2(ll, ~) + Z6~/2 n < y + 7e.I t r e m a i n s to show t h a t l(~)E :~'. W e can satisfy F ' - I as follows. Suppose t o ~ dora 1(8) a n d t o # 0,1. Consider the stopping times t 1 .. . . . tk on l(e) on ~ . F o r n sufficiently large t o is b e t w e e n two stopping times which correspond to stopping p o i n t s on a c o m m o n c o n t o u r c= of ~ n . T h u s we o b t a i n a sequence of contours {Ca) w i t h c, c ~ ' ~ . W e c a n n o t assert t h a t these contours t e n d to a single p o i n t of I~ r- W. H o w e v e r , t h e r e is a subsequenee which does h a v e a limit point, s a y ~-E l~ r - W. T h e corresponding s t o p p i n g times yield {r=}, {sn}. T h e cases t o = 0 , 1 can be handled similarly. T h e checking of F ' - 2 , F ' - 3 will be o m i t t e d .
Proo] o/ (c). I f a 1-chain l' E:~' can be e x t e n d e d continuously to [0,1] w i t h values in I~ t h e extension will a u t o m a t i c a l l y be a n arc in :~. F o r each ~ we consider a n n u l a r regions An~ a r o u n d each c o n t o u r of ~ n . W e show t h a t if no such annulus is crossed infinitely often b y l' t h e n l' can be e x t e n d e d continuously to [0, 1]. This will p r o v e (c) because the e x t r e m a l length of a f a m i l y of 1-chains, each 1-chain of which crosses some An~ infinitely often, is cr
G i v e n toe d o m l', t 0 # 0,1. L e t {rn}, {s~}, ~ be as in F ' - I . W e need o n l y p r o v e
lim l'(t) = ~ (t E d o m l'). (6)
t->to
2 5 6 A. MARDElg AND B. RODIN
L e t G be a neighborhood of ~e on IV. We wish to find a neighborhood of t o whose l'- image is in G, a n d for this we m a y assume G is a component of W - ~N for some h r. L e t A be the annular region around ~G chosen above; for definiteness assume A a n d G have intersection ~G. Now l'(r.), l'(s.)EG for sufficiently large n. If (6) failed we could find for some G, u n E d o m l ' with u . ~ t o and l'(u.)~GU A. A subsequence of {un} is monotonic so suppose u ~ T t o. Choose rt, and us,>ri, and ~ m , ~ SO t h a t rs,, us, Edoml'H~mv Since /'][~2m, e : ~ , there is a crossing of A during (rs,, u.). N e x t choose rs,, us,, s such t h a t rt, <us, < r . <u~ 2, a n d rs,, rs, E d o m l ' H ~ , . There m u s t be a crossing of A during (r~, us,). I n this w a y we see t h a t l' crosses A infinitely m a n y times. The cases t o =0,1 can be t r e a t e d similarly.
I I I . 3 . S o l u t i o n of t h e e x t r e m a l l e n g t h p r o b l e m s . The definitions of :~, :~* were given in 1.3, and du(z) =du(z; ~0, ~,/~, Y) was constructed in I L l .
THV.OREM III.3.1. Let W be an arbitrary open Riemann sur/ace and let du, ~, ~*
be defined with respect to an admissible partition of the ideal boundary of W. Then (a) ~(~) = IId~lt -~
(b) A(~*)= IId~ll ~-
Proo/ o/ (a). Using the notation of Chapter I I and replacing the pair F and ~ of I I I . 1 b y Fs and g2ns L e m m a I I I . l . 1 implies t h a t 2(F~)= Ildu~sit-2 Using o]dz I = Igrad Un] Idz]
~ n ( "
as an admissible metric for the problem ~(~.) (see I I I . 2 ) we find
and hence for all i,
II ~ , I1~ ~. < ~(Y,) < ~(F,) = II d~., II~,.
Consequently A(~,) = [[ du, [[~.
Since every curve in ~ contains a curve in ~n, A(:~) ~ ( ~ ) . On the other hand, f r o m L e m m a III.2.1
IId~ll-~=nm ~(:~)~>~(~).
The proof of (a) is now complete.Proo/ o/ (b). We write ~,~, F~ for ~ a n d F* of I I I . 1 . Since e v e r y curve in ~* contains a curve in F~ for all i we find
A *
a n d
I I~u~ll~
< (Y,). Conversely every level loci ua = c in ~ contains a curve in ~* a n d9 ~t *
application of the Schwarz inequality yields ~(~)~<
II ~u~ll~.
Hence ( ~ ) =II ~u~ll~..
Since every curve in 7 " is a curve in 7", ~ ( 7 " ) < ~ ( 7 " ) for all n. Therefore 2(7")~<
H du H 2. I t remains to prove t h a t H du II 2 <2(7*).
Set dun, = 0 in ~ n - ~ . , a n d consider the sequence of linear densities Qn, ldz [ = 1grad (un-un,)lidz [ in Un. According to L e m m a 1.2 there exists a subclass 7~* of 7*
with ~(7~*)=2(7*) and a sequence of numbers (i) such t h a t for all curves CEYn* , lim
f du * - du *, l < lim
~ [ dun - dun, l = O.i..--~oo t--.).~ j c
Here L ~ u : , = L o ~.~d n, = IIdun, ll~., as can be seen m o s t u* easily b y replacing c b y a simplicial a p p r o x i m a t i o n to c in a triangulation of W in which 8~n, is a cycle. We con- clude t h a t
fodu*
IIdunllL for all c E 7~*.F o r each c E~*, there exists a n u m b e r hr(c), depending on c, such t h a t c c l'ln for all n >~N(c), t h a t is c E :~* for all n ~> N(c). Otherwise c would have limit points on ~0, :t, or ft.
Since the countable union of classes of curves of infinite extremal length has infinite ex- tremal length, there exists a subclass :7'* of :~* with 2(:~'*)=2(:7*) such t h a t cE:7'* if and only if cE~7~* for all n>~N(c).
Set du n =0 in W - ~ n and a p p l y L e m m a 1.2 to the sequence of linear densities 9n I dz[ = 1grad (u - un)[ I dz]. There exists a subclass :7"* of :7 r* with ),(~"*) = 2(7'*) and a sequence of numbers {m} such t h a t for all curves cE:~"*,
12mlf du*-fodu*~l<lim~_,~ foe,,Id~l=O.
We h a v e shown above t h a t ~e du* = ]1 dum I1~ when m/> N(c). Again we m a y conclude t h a t ~c du* exists a n d t h a t
f c
du* =
IIdull 2 for all CE r,, 7Now using
e ldz
[= ]grad u I[dz
[ as an admissible linear density for 2(7"*), we findConsequently,
]ldul] ~= inf fcdu* <~ inf f ~Id~ J.
CE~ "* C E~"*
Ildu[[~
<2(T'*)=R(:~*), and the theorem is proved.258 A . M A R D E N A N D B . R O D I N
I I I . 4 .
S o m e consequences.
Properties of u can be discovered b y e x t r e m a l length considerations.COROLLARY III.4.1. (Uniqueness theorem/or u(z; ~o, g, fl, 7).) 1[ v is harmonic in W with []dv]] < [[du[[ and Scdv >~ l /or almost all cE:~, then v = u +a constant. The same con- clusion is true i[ [IdYll >~ [[du[[ and Srdv * >~ [[dv][ 2/or almost all c e ~ * .
Proo[. We will prove only the first statement; the proof of the second is similar. The hypotheses imply t h a t 01[dz[ = [grad v[ l dz[ is the e x t r e m a l metric for a subclass ~:1 of ff with ~ ( : ~ - ~ ) = co. We already know t h a t o[dz] -~ ]grad u[ [dz] is the e x t r e m a l metric for a subclass :~' of :~ with ~ ( ~ - ~ ' ) = oo. Setting :~2=:~1 N :~' we h a v e t h a t 2 ( ~ ) = 2 ( : ~ ) since :~ =:~2 O ( : ~ - ~ 1 ) U ( ~ - : ~ ' ) . Hence ol[dz[ and o[dz[ are b o t h e x t r e m a l metrics for the class ~ a n d thus 01-~ 0 or [ grad v [ - - [grad u [. This implies t h a t dv = (cos 0) du - (sin 0) du*
for some constant 0. L e t d be a cycle t h a t separates g0 from g, t h e n Sadu* = +_ ][du[[ ~.
Hence because So du = 1 and Se dv = 1 for almost all c e ~, 0 = 0.
COROLLARY 111.4.2. The Dirichlet integral [[du(z; o% o~, fl,
r)ll a)
increases (or is unchanged) when components in fl or 7 are placed in go, o~ or fl; b) decreases (or is unchanged) when components in g0, g or fl are Tlaced in fl or 7.Next, we will formulate the corresponding results for v(z; p, ~, fl, 7). L e t z be a local p a r a m e t e r a b o u t p, z(p) = 0, a n d
v(z; p, :r ~,
r) =log[z[
+e,(p, ~, ~,~,) +o(1)
be the expansion of v in z. F o r small r > 0 set g0 = { I z [ --- r } and let ~r be the family of curves :~----:~(~0, ~, fl, 7). Similarly, let :~* be the conjugate class of curves :~*=~*(~0, :r fl, 7).
An e l e m e n t a r y inequality gives for r' < r
~(~:r) ~>~(~,) + (log r/r')/2rc,
a n d hence 2zr2(~) + l o g r is increasing as r decreases and has a limit ~< + oo.
L~MMA III.4.3. 1 - c~(p, g, fl, 7) = lim 2~r2(~r) + log r.
r-*O
Proof. W e first consider the effect of changing the local p a r a m e t e r a b o u t p from z to w where w = a z + o ( z ) or log]w I = l o g [ z I + l o g ] a [ +o(1) (as z->0). Choose r 1 so t h a t the disk [w[ ~ r I is the largest such disk t h a t fits inside the disk
Iz[
~<r. T h e n2=2(~r) + l o g r <2~r2(~r,)+log r l - l o g [ a I +o(1) and lira (2~r~(~) + log r) ~< lira (2rr~(~,,) + log rx) - log
[a l"
r--~O rl--*0
B y considering the smallest disk Iw[ <r~ which contains ]z] ~<r the opposite inequality can be established and hence, using an obvious notation,
,-~01im (2~2~(~) + log~r) + log dzzdW ~ o = ~-.olim (2~2w(~) + log~r).
Now consider v =v(z; p, :r 8, 7) and let w be the local parameter w = C +~'* = ze ~p" ~" z. ~) + o(z),
where v* is the harmonic conjugate to v. Setting v, = (v - l o g r)/(1 - l o g r) and ~0 = { I w ] = r ) we see that v~ =u(w; ~0, ~, 8, ?) and hence ]]dvxH-~ =2w(:~). A simple computation gives ]]dvlJl~=(1-log r)-~lldv]l~=2g(1-log r) -1 where the Diriehlet integrals are extended over W - {Iw I ~< r}. Consequently
lim
(2a1~(y,)
+ log r) = 1r--~
and the validity of the lemma is clear.
COROLLARY III.4.4. (Uniqueness theorem/or v(z; p, ~, fl, ~).) Let w be harmonic in W - { p ) with the expansion w = l o g I z l § c + o(1) at p in the local parameter z (z(p) =0). Set A =lima~wSo~ wdw* where ( ~ ) is an exhaustion o/ W and set ~o(r)= (z : v(z)=log r).
I / /or some decreasing sequence r =r(n), lim r ( n ) = 0 , we have
S s d w > ~ l - l o g r, almost all sE:~r=:~(go(r), ~, 8, ~) and all r=r(n),
and c -c~(p, o:, 8, ~) >~ (A/2:~) - 1,
then w~--v(z; p, ~, fl, ~,) + a constant.
Proo/. We note the condition above is independent of the choice of local parameter z and t h a t if w - - v then A =2~.
Choose the local parameter $ = e x p (v + i v * ) = z exp cz +o(z) about p, where v* is the conjugate harmonic function to v. Let A be t h e disk I$[ < r and set w ' = w / ( 1 - l o g r) in
W - A . Corollary III.4.1 implies t h a t
IIdw'll =(1 -log r) lldwll-2
where the Dirichlet integrals are extended over W - A . We then find that, setting d = c - cz, 2~(1 - log r) ~ (A - 2rid - 2~ log r) -~ + log r < 2ze~(:~,) + log r.
As r =r(n) approaches 0 the left side decreases and the right side increases. We obtain on using the lemma that
2 + d - A / 2 ~ <. 1.
260 A . M A R D E N A N D B . R O D I N
B y assumption equality must hold. Hence 2~(:7r)=
Hdw'[[-~
for all r=r(n), and from Co- rollary III.4.1,w' =u((; ~o(r), ct, fl, 7) + a constant (depending on r).
B u t we have previously observed t h a t u ( ( ) = (v(z)-log r)/(1 - l o g r) and we conclude t h a t w--- v + a constant.
Remark. The hyptheses of Corollary III.4.4, which are satisfied b y w + a constant if they are satisfied b y w, can be replaced b y the following conditions. Set ~0'(r) = {z: w(z) = log r}. Then require
f s d W >/1 - log r , almost all e ~ = ~(0~(r), 8
8,
7)and c - c z ( p , ~, 8, 7) <~1 -A/2xt.
These conditions are not necessarily satisfied if a constant is added to w b u t it now follows, exactly as above, that w ~ v . Indeed, let ~ =exp(w +iw*) and w ' = ( w - l o g r ) / ( 1 - l o g r).
The lemma implies t h a t lim (2~t2(:~)+log r ) = l - c z + c and we conclude t h a t w ' = u(~; ~0, ~, 8, 7). But then (in W - A ) [[d(w-v)][z= - S o ~ ( w - v ) d ( w - v ) * =0.
COROLLARY III.4.5. Notation as in Corollary III.4.4. Assume f dw* >~ 2Jr, almost all E ~*, T
and c -c~(p, ~, 8, 7) >~ (A/2~t) - 1.
Then w----v(z; la, :r 8, 7) +a constant.
COROLLARY III.4.6. The constant cap, ~, 8, 7) a) decreases (or remains unchanged) when components in fl U 7 are placed in ~ U 8, b) increases (or remains unchanged) when com- ponents in ct U fl are placed in fl U 7.
COROLLARY III.4.7. a) I[ o~ o is a curve imbedded in asur[ace W,[[du(z; ~o, o~,fl, 7)[[ = 0 implies[[du(z; ct0, ct, 8, )')[[ = 0 / o r any other curve O~o; b) %(p, ct, f l , 7 ) = - c o [orsomepe W implies cz(p, ~, 8, 7) = - co/or all points 19 e W.
IV. Plane regions
I V . i . D e f i n i t i o n a n d e x t r e m a l p r o p e r t i o s of t h e slit m a p p i n g s . Assume t h a t W is a plane region and that % are each single ideal boundary components. Furthermore, we will assume t h a t
Ilau( ; c (p,
Then du(z; go, ~t, fl, 7)* has a single period equal to
Ildull
along dividing cycles separating~0 from ~, and dr(z; p, ~t, fl, ~,)* has a single period of 2n around cycles bounding a region containing p. Hence the functions
go, ;,)=expr2 (u +iu*)/lldull l(z; p, a, fl, 7)=exp(v +iv*)
are determined to a multiplicative factor k, I kl = 1. B y using a standard approximation argument, it is seen that/(z; go, a, fl, 7) and/(z; p, a, fl, ~,) are univalent.
De/inition IV.I.1. Assume A is a plane region with a partition (go, ~, fl, 7) of its boundary subject to the conditions of 1.3 and such t h a t go and ~ consist of single components.
If A is contained in the annulus 1 < Iz[ < R and u(z; go, a, fl, 7)-~log[z[/log R then A (or A(/~)) is referred to as an extremal slit annulus.
Assume A is a plane region with a partition (~, fl, 7) of its boundary subject to the conditions of II.4 and such that ~ consists of a single component. If A is contained in the unit disk [z[ <1 and contains the point z = 0 such t h a t v(z; O, ~, fl, 7)--logiz[ +1, then A is referred to as an extremal slit disk.
I t is easily seen t h a t if {Iz I = r } is contained in the interior of A, then A is an extremal slit disk if and only if i f3 {]z] >r} is an extremal slit annulus (after the mapping z~z/r).
Indeed the necessity was seen in II.4. Assume conversely t h a t A ' = A n {Iz[ > r } is an ex- tremal slit annulus (after the mapping z-+ z/r) with I z ] = r an isolated boundary component.
Set go= {1 1 =r}, 8, and
as given with A,u=u(z; go, ~, fl, 7)=loglz/rl/log r -1, and v=v(z; O, o~, fl, 7).
We find after using a suitable exhaustion t h a t
(u-v)d(u-v)*=
IId(u-v) lli.= -
H e n c e y., ~ v .
Using the preceding result as justification we will deal only with extremal slit annuli.
These can be characterized geometrically as follows.
THEOREm IV.1.2. Suppose A is a region contained in {1< Is] < R } with admissible boundary partition (go, ~, fl, ~'). Set ~ = ([ z [ log R) -1. The [ollowin9 conditions are equivalent.
(a) A is an extremal slit annulus A(R)
(b) Sc Q
Idzl/>
1 and Sa~ I dz ] >~ 2g/log R / o r almost all c E y, d E y*.(c) 2(y) ~< (log R)/2z~ and ~ce]dz I >1 1/or almost all c E ~.
(d) 2(~)/> (log R)/2rc and ~ ~ [dz [/> 2~t/log R for allmost all d E ~*.
17 - 6 6 2 9 4 5 Acta mathematlea. 115. I m p r i m 6 le 15 m a r s 1966.
