burt rodin- schwarz's lemma for circle packings

8/3/2019 Burt Rodin- Schwarz's lemma for circle packings

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Inve nt. m ath. 89,271-289(1987) I f l v e f l I i o f l ~mathematicae9 Springer-V erlag 1987

Sehwarz ' s l emma for c i rc le pack ingsB u r t R o d i n *Dep artment of M athematics, Un iversity of California, La Jolla, C A 92093, USA

1 . I n t r o d u c t i o n

C o n f o r m a l m a p p i n g s c a n b e a p p r o x i m a t e d b y c ir cle p a c k in g i s o m o r p h i s m s ;t h i s is p r o v e d i n [-1 0]. R o u g h l y s p e a k i n g ( s ee S e c t. 5 f o r a m o r e d e t a i l e dd e s c r i p t i o n ) , a b o u n d e d r e g i o n f2 is a l m o s t f i ll e d b y e - c ir c le s 7 f r o m t h e r e g u l a rh e x a g o n a l e - c ir cl e p a c k i n g o f t h e p l an e . D e n o t e t h i s a p p r o x i m a t e c i r cl e p a c k -i ng o f Q b y Q=. B y r e s u lt s o f A n d r e e v a n d T h u r s t o n , t h e r e is a c o m b i n a t o r i a l l yi s o m o r p h i c c ir c le p a c k i n g D + o f t h e u n i t d i s k D . D e n o t e t h is i s o m o r p h i s m ,s u i t a b l y n o r m a l i z e d , b y 7 ~--,7 ': f 2= --* D =. L e t f= b e t h e p i e c e w i s e l i n e a r q u a s i c o n -f o r m a l m a p p i n g d e t e r m i n e d b y t h e a s s o c ia t e d t r ia n g u l a ti o n s . A s e ~ 0 , f ~ c o n -v er ge s u n i fo r m l y o n c o m p a c t a t o t h e R i e m a n n m a p p i n g f u n c ti o n f : f 2 ~ D .

L e t H C P N b e N - g e n e r a t i o n s o f t h e r e g u l a r h e x a g o n a l c i rc l e p a c k i n g , a n d l etH C P } b e a c i r c l e p a c k i n g t h a t i s c o m b i n a t o r i a l l y i s o m o r p h i c t o H C P N. A s s u m et h a t D i s t h e s m a l l e s t d i s k c o n t a i n i n g H C P N , a n d t h a t I t C P } is c o n t a i n e d i n D .I n th i s c o n t e x t w e p r o v e t h e f o l l o w i n g a n a l o g o f t h e cl a ss i ca l l e m m a o fS c h w a r z . T h e r a d i i R o , R ; o f th e g e n e r a t i o n z e r o c i rc l e s o f H C PN a n d H C P }sa t i s fy R 'o 0 a s e , ~ 0 ; h e n c e t h e o p e n q u e s t i o n is e q u i v a l e n t t o : d o e sI~ ?fJO z[ - [ f ' l - ~ 0 ?T h e p r o o f o f T h e o r e m 5.1, t h e S c h w a r z l e m m a a n a l o g , m a k e s u se o f t hef a c t t h a t i n a n y H C P ~ , t h e a v e r a g e o f t h e r a d i i o f t h e c i rc l e s o f g e n e r a t i o n N isn e v e r s m a l l e r t h a t t h e r a d i u s o f t h e g e n e r a t i o n z e r o c i rc l e m u l t ip l i e d b y a

* Research supported in part by the NSF and DA RPA under the ACM P


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272 B. Ro dinp o s i t iv e a b s o l u t e c o n s t a n t (i.e., i n d e p e n d e n t o f N a n d t h e p a r t i c u l a r p a c k i n g ).T h i s r e s u lt , T h e o r e m 2 .2 , is p r o v e d a s a c o n s e q u e n c e o f d is c r et e p o t e n t i a lt h e o r y f o r h e x a g o n a l l a tt ic e s . I n t h e c o u r s e o f d e v e l o p i n g t h e n e c e s s a r y f u n -d a m e n t a l s w e p r o v e th e f o l lo w i n g r e s u lt ( T h e o r e m 3.2) o n t h e e x i s te n c e o f a" f u n d a m e n t a l p o t e n t i a l " : T h e r e i s a d i s c r e te r e a l v a l u e d f u n c t i o n 2 d ef i n ed o nt h e h e x a g o n a l l a t t i c e s u c h t h a t 2 i s h a r m o n i c ( i , e . , t h e v a l u e a t a l a t t i c e p o i n t i s

a v e r a g e o f t h e v a l u e s a t t h e s ix n e i g h b o r s ) e x c e p t a t 0 a n d 2 (c 0 =~ 2 -1 o gl c~ 1h e+ c o n s t . + O f o r a l l l a t t i c e p o i n t s cc

Acknowledgements. I would lik e to thank Stefan W arschawski for m any interesting discussions onthis material. While trying to prove Th eorem 2.2 I had interesting discussions with my colleagues,M ike Fredman and Janos Kom los. T he y found a combinatorial proof [6]. I subsequently foundthe potential theoretic proof given here.

2 . S u b h a r m o n i c i t y o f t h e r a d i iA circle packing is a c o l l e c t i o n o f c i rc l e s in t h e p l a n e w h o s e i n t e r i o r s a r ed i s j o i n t . T h e nerve o f t h e p a c k i n g i s t h e i m b e d d e d g r a p h w h o s e v e r t i c es a re t h ec e n t e r s o f t h e c i r c l e s ; th e l in e s e g m e n t j o i n i n g t w o v e r t i c e s i s a n e d g e i f a n do n l y i f t h e c o r r e s p o n d i n g c ir c le s a r e t a n g e n t . T w o c i rc l e p a c k i n g s a r e com-binatorially equivalent i f t h e i r n e r v e s a r e i s o m o r p h i c . T h e r e g u l a r h e x a g o n a lp a c k i n g o f t h e p l a n e b y c i rc le s o f e q u a l r a d i i is d e n o t e d b y HCP; i t s n e r v ed e t e r m i n e s a p a v i n g o f t h e p l a n e b y e q u i l a t e ra l t ri a n gl e s. W e l e t HCPu (N= 0 , 1 , 2 , . . . ) d e n o t e N g e n e r a t i o n s o f HCP s t a r t i n g f r o m s o m e b a s e c i r c l e .T h e r e a r e 6 N c i rc l es o f g e n e r a t i o n N i f N > 1. W e l e t HCP~ d e n o t e a p a c k i n gt h a t is c o m b i n a t o r i a l l y e q u i v a l e n t t o HCPN. ( C l e a r l y t h e c i r c l e s i n a n HCP~n e e d n o t b e o f e q u a l r ad ii . H o w e v e r , i f a p a c k i n g i s c o m b i n a t o r i a l l y e q u i v a l e n tt o a l l o f HCP then i t i s , in fac t , HCP [10 ] . )

A n HCP~ p a c k i n g c o n s i s t s o f a n i n n e r c i rc l e s u r r o u n d e d b y s ix ta n g e n tc i r c l e s ( F i g. 2 .1 ). T h e f o l l o w i n g r e s u l t f o r s u c h p a c k i n g s is s t a t e d i n [ 2 ; p . 5 7 6 ]( th e r e f e r e n c e fo r a p r o o f g iv e n t h e r e a p p e a r s t o b e i n c o r re c t ). F o r t h e c o n -v e n i e n c e o f th e r e a d e r w e g i ve a s e l f -c o n t a i n e d p r o o f h er e.

r2 h

Fig. 2 .1.


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Schwarz's lemm a for circle packings 273L e m m a 2 . 1 . I n a n y H C P ~ p a c k i n g o n e h a s(2.1) R o -


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2 7 4 B . R o d i n1 2 = t(2.5) I f ' (0) [ < 2 s ! I f (e~~ dO

w h i c h h o l d s f o r a f u n c t i o n f a n a l y t i c i n th e c l o s e d u n i t d i s k D . A s d e s c r i b e d inS e c t. 5 , i f f is a R i e m a n n m a p p i n g f u n c t i o n w e i m a g i n e f t r a n s f o r m i n g a ni n f in i t e si m a l c i r c l e p a c k i n g o f D ( th i s c o r r e s p o n d s t o HCP2v o n t o an i n f i n i t e s i -m a l c i rc le p a c k i n g o f f (D) ( t h i s c o r r e s p o n d s t o HCP~). W e i n t e r p r e t ] f ' ( z ) l a st h e r a t i o o f t h e r a d i i o f t h e i m a g e c i rc l e a n d t h e p r e i m a g e c ir cl e. T h u s (2 .5 )s a y s t h a t f o r t h e i n f i n i t e s i m a l p a c k i n g o f f(D), t h e a v e r a g e r a d i u s o f t h e o u t e rc i rc l e s d o m i n a t e s t h e r a d i u s o f t h e c e n t e r c ir c le . S i n c e t h e h a r m o n i c m e a nv a l u e f o r m u l a

I 2 ~(2.6 ) log [ / ' ( 0 ) l = ~ ! log [f'(e~~ dOh o l d s w i t h e q u a l it y , o n e m i g h t s u s p e c t t h a t a m o r e p r e c i se f o r m o f T h e o r e m2 . 2 c o u l d b e o b t a i n e d b y c h a n g i n g ( a m o n g o t h e r t h i n g s ) t h e a r i t h m e t i c m e a n t ot h e g e o m e t r i c m e a n .

3. D iscrete p otential theoryL e t og =e i~13 a n d l e t h > 0 . T h e s e t HL(h)={hm+hnod: m, neZ } i s ca l l ed thehexagonal lattice of mesh h. T h e neighbors o f ct~HL(h) a r e t h e s i x p o i n t s+ h cok, 0 _< k < 5. T h e l a t t i c e p o i n t ~ is o f g e n e r a t i o n < N i f t h e r e is a s eq u en ce0 = % , % . . . . . c~N=c~ suc h tha t c~j i s a ne ig hb or o f c~;_1 fo r I < j < N . W e l e tHL(h, N) d e n o t e t h e s u b s e t o f HL(h) c o n s i s t in g o f l a t ti c e p o i n t s o f g e n e r a t i o n=< N . T h e s e t o f la t t i c e p o i n t s o f g e n e r a t i o n e x a c t l y N is d e n o t e d OH L(h, N). I ft h e v a l u e o f h is u n d e r s t o o d w e o f t e n s im p l i f y t h e s y m b o l s H L(h), H L(h, N),OH L(h, N ) to H , H N, c~H .

I f u is a c o m p l e x v a l u e d f u n c t i o n d e f i n e d a t c ~ eH a n d i ts s ix n e i g h b o r s , t h e nt h e discrete Laplacian o f u a t a is d e f i n ed b y }3 . 1 ) D h u ( ~ ) = 3 ~ - k u(a+hcok)--6u(cO "T o m o t i v a t e ( 3.1 ), su p p o se U ( x , y ) is c l a ss C 4 o n t h e co n v e x h u l l o f t h e s i x

n e i g h b o r s o f c~. A p p l y T a y l o r ' s f o r m u l a i n t h e f o r m (~k = R e cok, t/k = I m cok):

hJ / 0 0 \ ~ h4 [~kAq_tlkL~ 4U (flk )(3.2) U(~x+hr L j[. k 4 k~ x+ q k~ ) U(~)+~. \Ox ay!= 0w h e r e flk l i e s o n t h e s e g m e n t j o i n i n g ~ a n d ~+hcok. A d d t h e s i x eq u a t i o n s ( 3 . 2 )fo r k = 0 , 1, . . ., 5. A f t e r s i m p l i f y i n g b y Y , 4 k = Z t / k = 0 , ~ , k - -= 2 4 3 42 2 3 4 k~k =0= ~ k t / k = ~ a t / k = ~ q k = 0 ' Y, 4 2= 9/4 , ~ r /k = 3 /4 , ~ 4 3 t / k = ~ 3o n e o b t a i n s( 3 . 3 ) L U (o~+ he)k)=6U (oO +~--(Uxx(a)+ Uyy(cO)+h4E(oO

k = 0


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Schw arz's lemm a for circle packings 275w h e r e IE (c0[ < 9 M 4 / 4 ! i f M 4 is an u p p e r b o u n d f o r t h e m a g n i t u d e o f t h e f o u r t ho r d e r p a r t i a l d e r i v a t i v e s o f U o n t h i s r e g i o n . T h u s t h e d i s c r e t e L a p l a c i a nd e f in e d b y (3 .1 ) a p p r o x i m a t e s t h e c la s s ic a l L a p l a c i a n t o s e c o n d o r d e r :

(3.4) ID h U(oO - A U ( c 0 [ < h 2 4 M= 4 "A r e a l - v a l u e d f u n c t i o n u d e f i n e d o n H N i s s a i d t o b e s u b h a r m o n i c o n H N i f

D h U ( e ) > O f o r c ~ e H N - c ? H N . S u c h f u n c t i o n s o b v i o u s l y o b e y t heM a x i m u m p r i n c i p l e . S u p p o s e u is s u b h a r m o n i c o n H N . I f u a t ta i n s i ts m a x i m u mo n H N - (? HN t h e n u i s c o n s t a n t o n H N .

T h e m a x i m u m p r i n c i p l e i m p l i e s t h e ex i s te n c e , a s w e l l a s u n i q u e n e s s , o fs o l u t i o n s t o D i r i c h le t 's p r o b l e m f o r P o i s s o n ' s e q u a t io n .Proposit ion 3.1 . S u p p o s e f : O H N - , I R a n d F : H N - - O H N - -~ ] R a r e g i ve n . T h e n t h e r eis a u ni qu e f u n c t io n u : H N ~ I R s uc h t h a t u = f o n ~ H N a n d D h U = F o n H N - O H N.P r o o f C o n s i d e r O h a s a li n e a r t r a n s f o r m a t i o n o f t h e f o l l o w i n g f i ni te d i m e n s i o n -a l r e a l v e c t o r s p a c e s : t h e d o m a i n s p a c e i s a l l r e a l - v a l u e d f u n c t i o n s o n H Nw h i c h v a n i s h o n 0 H N , a n d t h e r a n g e s p a c e is a ll re a l - v a l u e d f u n c t i o n o n H N _~ .T h e m a x i m u m p r in c ip l e s h o w s t h a t t h e n u ll s p ac e o f D h c o n s i s ts o f z e r o a l o n e .S i n c e t h e d o m a i n a n d r a n g e h a v e t h e s a m e d i m e n s i o n , D h i s s u r j e c t i v e a s w e l la s i n j e c t i v e .

T h e r e f o r e t h e r e is a f u n c t i o n u o o n H N w h i c h v a n i s h e s o n c~HN a n d s a t i s f i e sDhUo=F. L e t v b e t h e f u n c t i o n o n H N w h i c h is e q u a l t o f o n 0 H N a n d v a n i s h e se l se w h e r e . T h e n t h e r e is a f u n c t i o n v 0 w h i c h v a n i s h e s o n c? HN a n d s a t i s f i e sDhVo=DhV. T h e f u n c t i o n U = U o + V - V o i s t h e d e s i r e d s o l u t i o n . I t i s c l e a r l yu n i q u e .

L e t c ~ e H N a n d f l o r i N _ l , N > 2 . W e u s e P r o p o s i t i o n 3 . 1 t o d e f i n e t h e d i s c r e t eG r e e n ' s f u n c t i o n g N ( e , / ~ ) f o r H u a s f o l l o w s :(3 .5 ) g N (~ , f l ) = 0 i f ~ H N( 3 . 6 ) D h g N ( ~ , f l ) = l O / s i f ~ H N _ I - - { f l }

/ - ~ v ~ i f c r[ 3h 2w h e r e t h e L a p l a c i a n , a s i n (3 .6 ), is a l w a y s t a k e n w i t h r e s p e c t t o t h e fi rs tv a r i a b l e o f gN ( ' , " ) . T h e m a x i m u m p r i n ci p l e s h o w s t h a t gN (Ct, f l ) > 0 if ~ H N _ 1.T h e s y m m e t r y o f g ~ m a y b e d e d u c e d e as il y f r o m P r o p o s i t i o n 3.4.

W e w a n t t o p r o v e t h e e x i st e n ce o f a f u n c t i o n o n H L ( 1 ) w h i c h is h a r m o n i ca w a y f r o m t h e o r i g i n a n d w h i c h g r o w s l i k e ( 1 / 2 ~ z ) l o g I z l u p t o a n e r r o r t e r m o fo r d e r O ( 1 / I z l ) . W e h a v e b e e n u n a b l e t o f i n d th i s r e su l t i n t h e l it e r a tu r e . I n t h ec a s e o f t h e s q u a r e l a t t ic e t h e e x i st e n c e o f s u c h a f u n d a m e n t a l p o t e n t i a l w a sf ir st p r o v e d b y M c C r e a - W h i p p l e [ 9 ] . A d i ff e re n t p r o o f , b u t w i t h o u t t h e O ( I / I z l )e r r o r e s t i m a t e , a p p e a r s i n S p i t z e r [ 1 1 ] a n d V a n d e r P o l [-12 ]. O t h e r r e f er e n c e sa re : W a s o w [ 1 4 ] a n d F o r s y t h e - W a s o w [ 5 ] f or c o m m e n t s o n M c C r e a - W h i p p l e[ 9 ] , i n p a r t i c u l a r , t h a t [ 9 ] a c t u a l l y s h o w s t h a t t h e e r r o r e s t im a t e is O(1/ Iz12) ;D u f f i n [ 4 ] f o r t h e h i g h e r d i m e n s i o n a l c a s e w h i c h i s c o n s i d e r a b l y s im p l e r.


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276 B. Ro dinT h e o r e m 3 . 2 . 7here is a rea l -va lued func t ion 2 on the hexag onal la t ti ce H L ( 1 ) o fmesh 1 such tha t

( i ) D 1 4 ( o ) - 2 1 / 53 '(ii) D 1 2 ( c ~ ) = 0 i f c ~ e H L ( 1 ) - { 0 } ,

(iii) 2 ( ~ ) = 2 ~ l o g l c ~ l + c o n s t . + O ( i - ~ ) a s c~--,oe, c ~ e n z ( 1 ) .T h e p r o o f o f T h e o r e m 3 .2 1 w i ll re fe r to t h e f o ll o w i n g L e m m a s 3 .3 a n d 3 .4 :

L e m m a 3 . 3 . T h e dis cr ete L ap lac ian o f the fu n ct io n C~W-~eIR~'~:H L ( 1 ) ~ I I 7 is(3.7) D 1 e l R e ~ tz = --8-eiRerLY'S(z)3,where S(z) i s def ined by

9 2 x " z x + Y ] / ~ x - y V ~ ( z = x + i y ) .( 3 .8 ) S ( z ) = s i n ~ - + s i n ~- +sinZ ~ ,The func t ion S can be w r i t ten(3.9) S ( z ) = 3 1 z l 2 - ~ - ~ 8 1 z l" + E ( z )where E(z ) / I z [ 4 and i ts f i r s t order par t ia l der iva t i ves are un i formly bounded oncompact subse t s o f I~ .P r o o f o f L e m m a 3 . 3 . D i r e c t a p p l i c a t i o n o f D e f i n i ti o n 3 .1 g i v e s

(3 .10) 2/ ) l e = 3 e L k= 0 eS i R e c t e 1 " " 2 x " 2 x + y V ~ Y ]= - 3 e [ s m ~ - + s ln -~ + s i n 2 x - ~ ] / ~ ]

w h i c h p r o v e s ( 3 . 7 ) . A n o t h e r c o m p u t a t i o n s h o w s t h a tx+yv5 x-yv(3 .11) 2 S ( z ) = l - c o s x + l - c o s T t- l - c o s ~

2 x " ( x + y v S ) " ( x - y l / 3 ) 4= 2 + ~ ( x + y V ~ ) 2 + ~ ( x - y V ~ ) z 4 ! 4 ! 1 6 4 ! 1 6: 88 2 - 3 1 z l 4 + 2 E (z )

q-...

w h e r e(3.12) 2 E ( z ) = x 6 c ~ ( x ) + ( x + y V ~ ) 6 ( a ( x + y V ~ ) + ( x - y V ~ ) 6 ~ 9 ( x - y ] / ~ )I I t sh ou ld be remarked, al though this fa ct w il l not be used, that condi t ions (i i) and ( ii i) ofTheorem 3.2 determine 2 un iquely up to an additive constant


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S c h w a r z ' s l e m m a f o r c i r cl e p a c k i n g s 2 7 7

a n d ~b i s t h e e n t i r e f u n c t i o n1 t 2 t 4

4 ) ( 0 = ~ - ~ ~ lO ! . .. .O n e c a n c h e c k t h a t t h e f u n c t i o n s (ygS)6 tyvS)6

( X 2 + y 2 ) 2 ' ( X 2 + y 2 ) 2 ' ( X 2 + y 2 ) Za n d t h e i r f i r s t o r d e r p a r t i a l d e r i v a t i v e s a r e u n i f o r m l y b o u n d e d o n c o m p a c ts u b s e t s o f C . B y (3 .1 2), t h e s a m e is th e r e f o r e t r u e f o r E(z ) / l z] 4.L e m m a 3 . 4 . L e t f2 b e a c lo s e d p a r a l l e lo g r a m in t h e p l a n e. L e t f : f 2 ~ R b e o fu n i f o r m l y b o u n d e d v a r i a t io n o n O ; t h a t i s, t h e r e is a n M s u c h t h a t t h e v a r i a t io no f f , a s a f u n c t i o n o f o n e v a r ia b l e , a l o n g a n y v e r t ic a l o r h o r i z o n t a l l in e i n 12 isl e s s t h a n M . T h e n(3.13) ~S e iR e~ f ( x , y ) d x d y = O ( ~ )a s o t ~ , o ~ H L ( 1 ) .P r o o f o f L e m m a 3 . 4 . L e t ~ = m + n e i ~ / 3 = ] 2 + i v w h e r e # = m + ( n / 2 ) , v = n ] / ~ / 2 .B e c a u s e o f b o u n d e d v a r i a ti o n w e c a n i n te g r a te b y p a rt s a n d o b t a i n

x2(y) e iV Y [ xz(y) \t i u x x 2 ( y ) _ i u x f )iR e'e f ( x , y ) d x = . f ( z ) e Ix,(r) ~ e d xxl ( r ) l ]2 x , (y)o ( 1 )a nd the re fo re t he i n t e g ra l i n (3 . 13 ) i s O(1 / ]2 ) . S imi l a r ly , i t i s O ( 1 / v ) . T h e r e f o r e i ti s O(1/( I]21 + Iv l) ) = O ( I / 1 / ~ + v 2) = O(1/ [~1) .P r o o f o f T h e o r e m 3 . 2 . T h e p o t e n t i a l 2 f o r t h e u n i t h e x a g o n a l l a tt ic e H L ( 1 ) isd e f i n e d b y( 3.1 4 ) 2 ( ~ ) = t / ~ i ] ( 1 - e " m U + " V ) ) d u d v , ( e = m + n e i ' ~ /3 ) "

~ - ~ 4 s in 2 ~ + s m 2 ~ + s i n 2

N o t e t h a t t h e l in e a r c h a n g e o f v a r i a b l e su = x , v= 893 . 15)

t r a n s f o r m s ( 3 . 1 4 ) i n t o

8 ( 1 - - e I R e n e ) d x d y(3 .16) 2(~) = 4S (z )w h e r e O = { z = x + i y : - n < _ x < n , - 2 n - x < y l / ~ < 2 n - x } a n d S i s d e f i n e d b y(3.8).


8/19

278 B. Ro dinI n o r d e r t o d i s c u s s t h e c o n v e r g e n c e o f t h e i n te g r a l i n (3 .1 4) w e n o t e t h a t

I1 - e " m " + " " )l< l m u + m v ] < ] /m 2 + n Z l / ~ + v za n d t h a t s i n ( t/ 2 ) > t/rc f o r 0 < t < re. T h e s e e s t i m a t e s g i v e


9/19

Schw arz 's lem m a for circle packings 279T h i s e s t i m a t e , t o g e t h e r w i t h t h e f a c t t h a t E(z ) / l z ] 4 is b o u n d e d ( L e m m a 3 .3),s h o w s t h a t I z c o n v e r g e s .

N e x t w e s h o w t h a t I3=O(1/Io~l) b y a p p e a l i n g t o L e m m a 3 .4 . I n o r d e r t ov e r i f y t h a t t h e h y p o t h e s e s o f t h is l e m m a a r e s a t is f ie d i t s u ff ic e s t o s h o w t h a tt h e f u n c t i o n( 3 . 2 2 ) ( 1 / 1 6 ) I z l ' * - ( 8 / 3 ) E ( z ) Izl 2 i-6 3S ( z ) Izl 2 S ( z ) Izl 4 lis b o u n d e d a n d h a s b o u n d e d p a r t ia l d e r i v a t i v e s o n f2 . S i n ce E(z ) / I z l 4 h a s t h e s ep r o p e r t i e s , i t s u ff ic e s t o s h o w t h a t [z[Z/S(z) h a s t h e m . I n e q u a l i t y ( 3.2 1 ) s h o w st h a t I z l 2 / S ( z ) is b o u n d e d . I f w e t a k e p a r t i a l d e r i v a t i v e s o f

Izl 2 1S ( z ) 3 3 l z l 2 ' E ( z )

8 1 28 + ~ 5w e f in d t h a t t h e y w il l b e b o u n d e d p r o v i d e d E(z) / ]z l 2 h a s b o u n d e d p a r t i a ld e r i v a t i v e s , w h i c h is t r u e b e c a u s e i t h o l d s f o r e a c h f a c t o r o f I z l 2 ( E ( z ) / I z l %

W e n o w s h o w t h a t1 6 ~ O ( 1 )( 3 .2 3 ) I 1 - - ~ - - l o g Ic~l+ c o n s t . + ~ .

L e t A ~ f 2 b e t h e d i s k {Izl


10/19

2 8 0 B . R o d i n

(3.27) 3 2 ~ z 2 ( 1 )T 2 (a ) = I 1 + c o n s t . + O

( ~ ) ~ , ~ , ~ , . . . 0 ~ ~ ~ c o s ~= c o n s t . + O 1 + ~ ~ - - d O + J d O0 1 b/ 0 tot[ ~ cosO U( 1 ) 1 6 r e ~ 2 ~/z= c o n s t . + O ~-~ + ~ - - l o g t ~ l + Z f ~ ~ c ~

3 o [ ~ l n c o s 0 U

T h e p r o o f w i ll b e c o m p l e t e i f w e s h o w t h a t~ ; ~ ~ o s u ~ u ~ 0 : o ( 1 )0 p cosO U

as p --+ o(3

o r e q u i v a l e n t l y , b y t h e s u b s t i t u t i o n t = p c o s 0 , t h a t

(3.28) 5 J" c o s u = 0o t u r _ t 2 "

S p l i t t h e a b o v e i n t e g r a l i n t oo b v i o u s e s t i m a t e

l o 0 p o oa s u m I f + I I ' I n t h e f ir s t i n t e g r a l u s e t h e0 t i t

I~c~ u = < l ~~3w h e r e k = ~ c o s u ( d u / u ) , a n d o b t a i ni

(3.29) j ' co su , u < log t / p 2 _ t 2

O < t < l

~ ( ; 1 (1 1- ~ - k ~ d s in - i t = 0 .0 \I n t h e s e c o n d i n t e g r a l w e u s e t h e e s t i m a t e

~ c o s u d U - s i n t O ( 1 )(3.30) I- ~ T -t t t tw h i c h f o ll o w s f ro m i n t e g r a t i o n - b y - p a r t s :

oo c o s u d ut UT h u s

( 3.3 1 ) S c o s u d u1 t u

s ~ n u ) ~ ~ - s in ~ O i l /- + s in - - = ~- t ~ .U t t t U 2 td t P. - s i n t d t + ! O ( 1 ) d t

~r - - ! ~ ~r ~ ~ _ ~


11/19

Schwarz's lem m a for circle packings 281T h e s e c o n d i n t e g r a l is c l e a r ly f in i te a n d t h e f ir st c a n b e e s t i m a t e d b y

P s i n t d t( 3 . 3 2 ) ! t / p 1 ~ s i n ( p s ) d sJP l i p S r 2I ~ s i n ( p s ) s in s ) d s + l f s ln (pS) ds

P 1/o \ s ] , / l - - s 2 P 1/o s1 i s i n ( p s ) s d s 1 s i n t d t ( 1 )

= P l / o r 1 6 2 -k--ipl t = 0 .T h i s c o m p l e t e s t h e p r o o f o f (3 .2 8) a n d , c o n s e q u e n t l y , T h e o r e m 3 .2 .

T h e G r ee n ' s f u n c t i o n g N (e t, fl) f o r H L ( h , N ) was def ined by (3 .5 ) and (3 .6 ) .W e s h a l l n e e d t h e f o l l o w i n g u p p e r e s t i m a t e f o r g N ( a , 0 ) a s a v a r i e s o v e r t h el a tt ic e p o i n t s o f g e n e r a t i o n N - 1.P r o p o s i t i o n 3.3. There i s an abso lu te cons tan t b such tha t fo r a l l N>= 2, i f~ e a H L ( h , N - 1) t hen 1(3.33) gN(ct, 0) < N 'P r o o f W e m a y a s s u m e u n i t m e s h s i z e h = l . I n p r o v i n g ( 3 . 3 3 ) w e m ay a l soa s s u m e t h a t ~ li es i n th e u p p e r h o r i z o n t a l e d g e o f H L ( 1 , N - 1), t h a t is, a m o n gt h e p o i n t s ( r eca l l co = e ~/3)(3.34) E N = { ( N - - 1 ) o , ( N - 1 ) o - I , ( N - 1 ) o - 2 , . . . , ( N - D o - ( N - 1 ) } .T h e m e t h o d w e a r e g o i n g t o u s e r e q u i r e s s p e c i a l c a r e a t c o r n e r p o i n t s . F o rt h a t r e a s o n w e r e p l a c e H L ( 1 , N ) b y a l a r g e r c o n f i g u r a t i o n / t N o b t a i n e d f r o mH L ( 1 , 2 N ) b y r e m o v i n g t h e u p p e r N r o w s o f l a tt ic e p o i n t s a n d t h e l o w e r Nr o w s o f l a t t i c e p o i n t s ( see F i g . 3 .1 ). W e d e f i n e an d co n s t r u c t t h e d i s c r e t eG r ee n ' s f u n c t i o n ~ N (~ ,0 ) f o r / ~ N , j u s t a s w as d o n e i n t h e ca se o f H N . T h em a x i m u m P r i n c i p l e i m p l i e s t h a t g N(a, 0 ) < ~ N ( ~ , 0 ) fo r ~ H N. T h e r e f o r e i t s uf -f i c e s t o p r o v e

.

Fig. 3.1.


12/19

282 B. Rodi n

0 < co n s t .(3.35) ~N(~, ) = ~ -f o r ~ e E N .

L e t u s sh o w th a t i n eq u a l i ty (3 .3 5) h o ld s i f t h e d i s c re t e o b jec t s t q u , gN a rerep lace d b y an a lo g o u s n o n d i sc re t e o b jec t s 12, G . T h a t i s, l e t f2 b e th e co n v e xh u l l o f / 1 N , a r e g i o n w i t h p o l y g o n a l b o r d e r . L e t G(z, 0 ) b e th e c l a s s i ca l Green ' sf u n c t i o n f o r O w i t h p o l e a t 0 . W e w i s h t o s h o w t h a t

co n s t .(3.36) G(a , O) < - - Nif a l ie s o n th e co n v e x h u l l o f E N .L e t z~---~(9(z/N) b e t h e R i e m a n n m a p p i n g f u n c t i o n o f f2 o n t o t h e u n i t d is k ,n o r m a l i z e d t o s e n d t h e o r i g i n t o t h e o r i gi n . T h a t is, q5 is th e R i e m a n n m a p o fth e r eg io n 1 2x o b t a i n e d f r o m f2 b y h o m o t h e t i c a l l y s h r in k i n g i t t o d i a m e t e r 4.L e t M b e th e m ax im u m o f IqS'(ff)[ a s N ~ v a r i e s o v e r t h e co n v ex h u l l o fH L ( 1 , N) . T h e r e f l ec t io n p r in c ip l e sh o w s th a t M is f in i te .L e t a b e a p o i n t o n t h e c o n v e x h u l l o f E N . L e t z 0 b e a p o i n t o n t h e u p p e re d g e o f t h e c o n v e x h u l l o f H L ( 1 , N ) su ch th a t I zo - a [ < 1 . T h e n

a lN ~ ) ( ~ ) d ~ 5 ~ "W ith th e h e lp o f (3.3 7) we o b ta in , f o r N su f f i c i en t ly l a rg e ,

~ ( ~ ) ~ - ~ o g ~-- 2 MG ( a , 0 ) = - l o g 1 -


13/19

Schwarz's lem m a for circle packings 283

- - l o g - - < - -(3.41) log (c t ) - Pa 2~ Iz l N '2 o (~) - ~ log < con st .(3.42 ) 1 I~l N

E s t i m a t e (3 .4 0) fo l l o w s f r o m t h e M a x i m u m P r i n c i p l e s in c e th e f u n c t i o n o n1t he le ft is h a r m o n i c f or ~ / ~ N - O / 4 s a n d is e q u a l t o 2 0 ( / ~ ) - ~ l o g l / 3 l f or

/3e~ H N. T he se b o u n d ar y va lu es a r e O(1/[ fll) by (3 .39), a nd 1 /31=O(N) fo r/ 3 e e / ~ N .

E st im ate (3 .42) fo l low s f rom (3 .39) s ince Ice = O (N ) fo r s e E N.E s t i m a t e ( 3 . 4 1 ) i s , f r o m a c e r t a i n v i e w p o i n t , a f u n d a m e n t a l p r o b l e m o fn u m e r i c a l a n a l y s i s : t o e s t i m a t e t h e e r r o r b e t w e e n a c l a s s i c a l s o l u t i o n t o

D i r i c h l e t ' s p r o b l e m a n d i ts f in i te d i ff e r e n c e a n a l o g u e . S u c h e r r o r e s ti m a t e s a r ee a s y t o d e r iv e i f t h e b o u n d a r y v a l u e f u n c t io n a n d t h e b o u n d a r y o f t h e r e g io na r e b o t h s u ff ic ie n tl y s m o o t h . T h e c a s e o f n o n s m o o t h b o u n d a r y v a lu e s a n db o u n d a r i e s w a s in v e s t i g a te d i n L a a s o n e n [ 8 ]. T h e f o l lo w i n g is a c o n s e q u e n c eo f L a a s o n e n ' s d i sc u s si o n. I f a region g2' has a piec ew ise smooth boundary, and i fthe in ter ior angles a t a l l corners o f ~ ' are less than ~z, then the e r r o r e h ( Z )be tween the so lu t ion to the c lass ica l Dir ich le t prob lem wi th ana ly t ic boundaryvalues and i ts f i n i t e d if ference analogu e fo r the square grid la t t ice o f mesh hsat is f ies eh (Z)=O (h ) as h-~O, un i form ly on co m pact subse ts o f ( 2 '~ c~g2'-{cornerpoints of c~Y2'}.

O n l y t w o m o d i f i c a t i o n s t o L a a s o n e n ' s a r g u m e n t [ 8 ] a r e n e e d e d t o m a k et h e a b o v e s t a t e m e n t a p p l y t o t h e h e x a g o n a l g r id o f m e s h h ; b o t h m o d i f ic a t i o n sa r e c o m p l e t e l y t ri v i a l . Th e f ir s t o n e is t o r e p l a c e Eq . ( 1) o f [ 8 ] w i t h Eq s . (3 .3 )a n d (3 .4 ) o f t h e p r e s e n t p a p e r . T h e s e c o n d m o d i f i c a t i o n is t o v e r i f y t h a tDh[z- -Zo[2=4, w h e t h e r t h i s d i s c r e t e L a p l a c i a n r e f e r s t o t h e h e x a g o n a l o r t h es q u a r e g r i d .T o p r o v e (3 .4 1) w e re p l a c e O a n d /4N b y t h e i r i m a g e s g 2' a n d / t ~ u n d e r t h eh o m o t h e t i c c o n t r a c t i o n z~--~z/N. L e t P a', Pc' b e t h e o p e r a t o r s w h i c h a s s i g n t o af u n c t i o n o n 0 t 2 ' t h e s o l u t i o n t o t h e d i s c r e t e ( r e sp e c t i v e l y c l a s si c a l) D i r i c h l e tp r o b l e m f o r O ' ( re sp e c ti v e ly / q ~v ). L a a so n e n ' s r e su l t g i v e s

1 3 4 3 1 ] K _ - ow h e r e t h e s u p r e m u m n o r m is o v e r t h e s u b s e t K c (2 ' c o n s i s t in g o f a ll z / N s u c ht h a t z ~ H L ( 1 , N ); (3 .4 3) is e q u i v a l e n t t o ( 3.4 1). Th i s c o m p l e t e s t h e p r o o f o fP r o p o s i t i o n 3 . 3 .

T h e n e x t re s u l t is a n a n a l o g o f G r e e n ' s s e c o n d i d e n t i t y . W e f o l lo w t h et r e a t m e n t i n D u f f i n [ 4 -] w h i c h r e q u i r e s o n l y s u p e r f ic i a l m o d i f i c a t i o n s t o a p p l yt o t h e h e x a g o n a l l a t t i c e .P r o p o s i t i o n 3 .4 . L et u , v be func t ions de f ined on HN, the f i r s t N genera t ions o fthe hexagonal la t t ice o f mesh h . Then


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284 B. Ro din

2 ~ v (a )u ( f l ) -u (cO v( f l )3.44) ,~n~ - , v ( ~ ) D h u ( ~ ) - u ( a ) D h v ( ~ ) = ~ ( , , O )

where the second sum is over a l l pa irs (~, fl) such tha t ~eO H u_ 1, f leO H u and ~ i sa neighbor of f t .P r o o f S e t u ' ( a ) = u ( a ) f o r ~ e H u _ 1 a n d u ' ( a ) = O f o r a ll o t h e r p o i n ts ~ in H , t h ei n fi n it e h e x a g o n a l l a t t i c e o f m e s h h . D e f i n e v ' si m i l ar l y . T h e n

V'(a)DhU ( a ) = ~ . v ( a ) - U ~ u ' ( a + h ~ ~ e H a e H " ' ~ k = 0

= ~ u ( 7 ) ~ v ' (7 - h c o * ) - 6 v '( 7 )7E H k = 0 != ~,, u'(7) Dh v'(y).

7 e HT h u s(3.45) if(c0 D h u (o - u (c0 D h v'(a) = O.

aeH

G r o u p t h e n o n z e r o t e r m s o f (3.45 ) i n t o t w o p a r t s :(3.46) ~ v ' (a)Dhu' (a)--u ' (~)DhV'(7)+ ~ V' (OODhU'(a)--U'(a)DhV'(~)=O.

c t e H N - I ~ 6 H N - 2

2For c~ec3Hu_1 w e h a v e D h U ' ( o O = D h U ( O O - - ~ ZU ( f l ) w h e r e t h e s u m i s o v e r t h ep o i n t s f le O H u w h i c h a r e n e i g h b o r s o f a . A s i m i l a r f o r m u l a h o l d s f o r v'. T h esec o n d su m m at i o n i n (3 .4 6) c an b e r e w r i t t en w i t h u , v i n s t ead o f u', v '. T h eseo b se r v a t i o n s t r an s f o r m ( 3 . 4 6 ) i n t o ( 3 . 4 4 ) a s d e s i r ed .

W e a r e n o w a b l e t o p r o v e t h e m a i n r e s u l t w e n e e d f o r t h is s e c ti o n.T h e o r e m 3 . 5 . There i s an abso lu te con s tan t c w i th the fo l low ing proper ty . L e t ube a pos i t ive subharmonic fun c t io n on HN, the f i r s t N genera t ions o f theh exa g o n a l la t t i c e o f mesh h . L e t ~ H N = {fix, f12 . . . . , fl6N} . Then(3.47) CU ( 0 ) ~ _ ~ - ( U ( f l l ) -'~ U ( f l 2 ) " - [ - . . + U ( f l6 N ) ) .P r o o f A p p l y P r o p o s i t i o n 3 .4 w i th v ( a ) = g u ( ~ , 0 ) w h e r e gN is t h e d i s c r e t eG r e e n ' s f u n c t i o n f o r H N d e f i n e d i n ( 3.5 ) a n d ( 3.6 ); o n e o b t a i n s(3.48) . . 2 ] / ~ 2( s o m e t h i n g p o s l t w e ) + ~ - u (0 ) = 3 ~ ~ (,~ p)g u(a, 0 ) u(fl).I n t h e s u m m a t i o n a b o v e t h e r e a r e a t m o s t t w o a ' s f o r e a c h f l e O H u , a n dP r o p o s i t i o n 3 . 3 g i v e s a n e s t i m a t e b / N f o r g N ( ~ ,0 ) s i n ce ae aH N _ 1. I n t h i s w aywe a re l ed to (3 .47) .


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Schwarz's lem m a for circle packings 2854 . P r o o f o f T h e o r e m 2 . 2W e a r e c o n s i d e r i n g a c i r c l e p a c k i n g HCP/v w h i c h i s c o m b i n a t o r i a l l y e q u i v a l e n tt o N g e n e r a t i o n s o f t h e r e g u l a r h e x a g o n a l p a c k i n g o f c ir c le s o f e q u a l r ad ii .A f t e r a s s o c i a t i n g t h e l at ti c e p o i n t 0 s i l L ( l , N ) w i t h t h e g e n e r a t i o n 0 ci rc l e o fH C P } a n d a r b i t r a r i l y a s s i g n i n g t h e l a t ti c e p o i n t I ~ H L ( 1 , N ) t o a f ir s t g e n e r a -t i o n c i r c l e o f H C P } w e o b t a i n a n a t u r a l b i j e c t i o n o f H L ( 1 , N ) w i t h H C P } . L e tr (~ ) b e t h e r a d i u s o f t h e c i r c l e i n H C P } a s s o c i a t e d t o t h e l a t t i c e p o i n tc ~ H L ( 1 , N ). B y L e m m a 2 . 1 , r : H L ( 1 , N ) ~ I ( i s a d i s c r e t e s u b h a r m o n i c f u n c -t io n . N o w a p p l y T h e o r e m 3.5 a n d o b t a i n t h e d e s i re d i n e q u a l i t y (2.4).

5 . S c h w a r z ' s l e m m a f o r c ir c le p a c k i n g sA c o n f o r m a l m a p p i n g o f a d i s k i n to i ts e lf h a s a d e r i v a t i v e a t t h e c e n t e r w h i c his n o g r e a t e r t h a n o n e i n m o d u l u s . T h e o r e m 2 .2 a ll o w s u s t o p r o v e t h ef o l l o w i n g a n a l o g o u s r e s u l t f o r c i r c l e p a c k i n g i s o m o r p h i s m s .T h e o r e m 5.1. There is an absolute constant a w ith the fol low ing property. L etHCPu be N generations of the regular hexagonal circle packing. Let D be thesmallest disk which contains HCPN. Let HCP} be any c irc le packing com-binatorially equivalent to H C PN an d also contained in D. T hen( 5 .1 ) R 'o < = a R owhere R o and R 'o are the generation zero circles in H C PN and HCP/v.P r o o f Let rk i , 1


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286 B. Ro dinp h i s m s . W e a r e g i v e n a b o u n d e d r e g i o n sQ i n t h e p l a n e a n d p o i n t s z 0 , z 1 i n ~Q.F o r e a c h s > 0 , f o r m t h e c i r c le p a c k i n g ~Q~ d e f i n e d a s f o ll o w s . C o n s i d e r t h er e g u l a r h e x a g o n a l c i rc l e p a c k i n g o f t h e p l a n e b y c i rc l es o f r a d i u s e a n d l e t H ~c o n s i s t o f t h o s e c i r c l e s c o n t a i n e d i n f2. L e t I s , t h e s e t o f i n n e r c i r c l e s, c o n s i s t o fa l l c i r c l e s 7 s u c h t h a t 7 i s t h e l a s t t e r m i n a f i n i te s e q u e n c e o f c i r c l e s w i t h t h ep r o p e r t i e s : (i) e a c h c i r c l e i n t h e s e q u e n c e b e l o n g s t o H e , (ii) t h e s i x H e-n e i g h b o r s o f e a c h c i r c l e i n t h e s e q u e n c e a l s o b e l o n g s t o H e, (iii) e a c h c i r c l e int h e s e q u e n c e i s t a n g e n t t o t h e p r e c e d i n g c ir c le , a n d (i v) t h e f l o w e r o f t h e i n i t ia lc i r c le i n t h e s e q u e n c e c o n t a i n s z o (t h e f l o w e r o f a c i r c l e in H ~ o r a c o m -b i n a t o r i a l l y e q u i v a l e n t c i r c l e p a c k i n g i s t h e c l o s e d c o n n e c t e d s e t c o n s i s t i n g o ft h a t c i r c l e a n d i t s i n t e r i o r , t h e s i x n e i g h b o r i n g c i r c l e s a n d t h e i r i n t e r i o r s , a n dt h e s i x t r i a n g u l a r i n t e r s t i c e s s o f o r m e d ) .

Le t B~ , the s e t o f border circles, c o n s i s t o f a ll c i r c le s f r o m H e w h i c h a r e n o ti n n e r c i r c l e s b u t w h i c h a r e t a n g e n t t o i n n e r c i r c l e s . T h e b o r d e r c i r c l e s a r ec o n t a i n e d i n ~ a n d c a n b e a r r a n g e d i n a s e q u e n c e s u c h t h a t e a c h o n e ist a n g e n t t o i t s p r e d e s s e s o r , a n d t h e f i r s t i s t a n g e n t t o t h e l a s t . T h e p o l y g o n a ll in e j o i n i n g s u c c e s s i v e c e n t e r s o f th e c i rc l e s in t h i s border cyc le i s a J o r d a nc u r v e w h i c h s u r r o u n d s a ll t h e i n n e r c i rc l e s I~ . D e f i n e f2+ t o b e t h e u n i o n o f I~a n d B ~. W e s h a l l r e f e r to g?e a s t h e e - c ir c le p a c k i n g a p p r o x i m a t i o n t o g2e (wi thd i s t i n g u i s h e d p o i n t Z o ) .

B y A n d r e e v ' s t h e o r e m [ 1 ] ( s e e a l s o [ 1 3 ] ) t h e r e i s a c i r c l e p a c k i n g O ~c o n t a i n e d i n t h e u n it d i s k D w h i c h i s c o m b i n a t o r i a l l y e q u i v a l e n t t o f ~ a n ds u c h t h a t a ll t h e c i r c le s o f D e w h i c h c o r r e s p o n d t o b o r d e r c i rc l e s o f ~Q~ a r et a n g e n t t o t h e c i r c u m f e r e n c e o f D . W . T h u r s t o n c o n j e c t u r e d t h a t t h is c i rc lec o r r e s p o n d e n c e ~ 2e ~D ~ , s u i ta b l y n o r m a l i z e d , c o n v e r g e s t o t he R i e m a n n m a p o f

o n t o D a s s ~ 0 . B y su i t a b l y n o rm a l i z e d w e s h a l l a l w a y s m e a n t h e f o l l o w i n g .P e r f o r m a M o b i u s t r a n s f o r m a t i o n f ix i ng D s o t h a t a ci rc l e o f f2e w h o s e f lo w e rc o n t a i n s z 0 c o r r e s p o n d s t o a c i r c le o f D e w h o s e f l o w e r c o n t a i n s t h e o r i g in .T h e n p e r f o r m a r o t a t i o n o f D s o t h a t a c ir c l e o f ~2~ w h o s e f l o w e r c o n t a i n s ap r e s c r i b e d p o i n t z 1 o f ~2 w i ll c o r r e s p o n d t o a c i r c l e o f D~ w h i c h l ie s o n t h ep o s i t i v e r e a l r a d i u s o f D .

T h i s c o n j e c t u r e i s p r o v e d i n R o d i n - S u l l i v a n [ 1 0 ] . I t i s s h o w n t h e r e t h a t a ss ~ 0 , t h e p ie c ew i se l i n ea r m a p p i n g s f~ c o n v e rg e to t h e R i e m a n n m a p f : f ~ D ,w h e r e f~ is t h e s i m p l i c ia l m a p o f t h e t r i a n g u l a t e d r e g i o n s d e t e r m i n e d b y t h ec a n o n i c a l i m b e d d i n g s o f th e n e r v e s o f (2~ a n d D ~. T h e n o r m a l i z a t i o n o f D ed e s c r i b e d a b o v e i m p l i e s t h a t f is n o r m a l i z e d b y f ( z o ) = 0 a n d R e f ( z l ) > 0.

C o n s i d e r a g a i n t h e c i r c le p a c k i n g i s o m o r p h i s m ? ~ + ? ' o f ~ 2+ ---,D e; h e r e a se l s e w h e r e w e i n t e n d t h a t D~ is t o b e s u i t a b l y n o r m a l i z e d . I t w i ll b e v e r y u s e f u lt o k n o w t h a t t h e m a p 7 ~ - + r a d ? ' / r a d ? is u n i f o r m l y b o u n d e d a b o v e ( in d e-p e n d e n t l y o f ~) o n c o m p a c t s u b s e t s o f Q .T h e o r e m 6 . 2 . L e t K b e a c o m p a c t su b se t o f Q . T h e re i s a c o n s t a n t M K w i t h t h ef o l l o w i n g p r o p er ty . L e t s > 0 be su f f i c i en t l y smal l and l e t ~-~7 ' be the c i rc l ep a c k i n g i so m o rp h ism o f a n e - ci rc le p a c k i n g a p p ro x i m a t i o n ~2e o f f2 on to asu i tab ly norm al i zed c i rc l e pac k ing D~ o f the un i t d i sk D . Th en r a d 7 ' / r ad 7_-


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Schwarz's lemmafor circle packings 287P r o o f F o r s u f f i c i e n t l y s m a l l e > 0 , c o n s i d e r a c i r c l e 7 i n f2~ s u c h t h a t 7 c~ K 4 rL e t N b e m a x i m a l w i t h r e s p e c t t o t h e p r o p e r t y t h a t f2~ c o n t a i n s a n H C PNc o n f i g u r a t i o n c e n t e r e d a t 7 . L e t A b e t h e s m a l l e s t d i s k c o n t a i n i n g t h i s H C PN.T h e i s o m o r p h i s m Y2~-~D~ m a k e s t h i s H C PN c o r r e s p o n d t o a n H C P/ ~ i n t h e u n i td i s k D . I f w e r e s c a l e t h e u n i t d i s k t o a d i s k t h e s iz e o f A w e m a y a p p l yT h e o r e m 6.1, t he S c h w a r z l e m m a a n a lo g , t o o b t a i n(6.2) 2 ra d 7 ' < a ra d 7w h e r e 2 is t h e s c a l i n g f a c t o r , 2 = t a d A . W e h a v e

89 (K , - f2) _< 2 _< d is t (K , ~; - f2)a n d t h e r e f o r e (6.2) g iv e s a n u p p e r b o u n d i n d e p e n d e n t o f ~ o f th e d e s i r e d t y p e :

t a d 7 ' 2 a


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2 8 8 B . R o d i n

C h o o s e N t o b e m a x i m a l w i t h r e s p e c t t o t h e p r o p e r t y t h a t t h e r e i s a nH C PN co n f i g u r a t i o n c o n t a i n ed i n f2~ an d cen t e r ed a t ~. I f w e r e s t r i c t 7 t oi n t e r s e c t a f i x e d c o m p a c t s u b s e t K o f f2 t h e n N - - , o o a s e--* 0. B y t h e H e x a g o n a lP a c k in g L e m m a o f [9 ] w e h a v e ~) /r l=l+O(sN) , ( j = 2 , 3 ) , w h e re s N ~ 0 a sN ~ o o . I f w e u s e t h is O(sN) n o t a t i o n a n d t h e f a c t t h a t rl/e


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S c h w a r z 's l e m m a f o r c i r cl e p a c k i n g s 2 8 9

References

1 . Andreev , E .M. : Convex po lyhedra o f f in i te vo lume in Lobacevsk i i space . Mat . Sb . , Nor . Ser . 83 ,256-260 (1970) (Russ ian) ; Math . USSR, Sb . 12 , 255-259 (1970) (Engl i sh)2 . B/t rf i, y , I ., F t i red i , Z ., Pac h , J . : Disc re te co nvex func t ion s an d pr oo f o f the s ix c i rc le con jec tureof Fe jes T6th . Can . J . Math . 36 , 569-576 (1984)3 . D o y l e , P . G ., S n e l l, J . L .: R a n d o m w a l k s a n d e l ec t ri c n e t w o r k s . C a r u s M a t h . M o n o g r . 2 2, T h eM a t h e m a t i c a l A s s o c i a t i o n o f A m e r i c a , ( 1 9 8 4 )

4 . Duff in , R . J . : Disc re te po ten t ia l theory . Duke Math . J . 20 , 233-251 (1953)5 . F o r s y t h e , G . , W a s o w , W . : F i n i t e d i f f e re n c e m e t h o d s f o r p a r t i a l d i f f e re n t i al e q u a t i o n s . N e w

Y o r k , London, S y d n e y : J o h n W i l e y a n d S o n s , I n c . 1 9 6 06 . F r e d m a n , M . , K o m l o s , J . : O r a l c o m m u n i c a t i o n7 . Ha rdy , G .H. , L i t t lewo od , J.E ., Po lyg , G . : Ineq ua l i t i es . Ca m br id ge U nivers i ty Press, 19528 . L a a s o n e n , P . : O n t h e d e g r e e o f c o n v e r g e n c e o f d i s c r e t e a p p r o x i m a t i o n s f o r th e s o l u t i o n s o f t h eDir ich le t p rob lem. Ann . Acad . Sc i . Fenn . , Ser . A . I . 246 , 1 -19 (1957)9 . M c C r e a , W . H ., W h i p p l e , F .J .W . : R a n d o m p a t h s i n t w o a n d t h r e e d i m e n s i o n s . P r o c . R . So c.

Ed inb. , Sect . A 60, 281-2 98 (1939-1940)1 0. R o d i n , B ., S u l li v a n , D . : T h e c o n v e r g e n c e o f c i rc l e p a c k i n g s t o t h e R i e m a n n m a p p i n g . J . D i ff er .G e o m . ( t o a p p e ar )1 1. S p i t z e r, F . : P r i n c i p l e s o f r a n d o m w a l k , 2 r id e d i t i o n . B e r l i n - H e i d e l b e r g - N e w Y o r k : S p r i n g e r19761 2. B a l t h , V a n d e r P o l : T h e f i n it e d if f er e n c e a n a l o g o f t h e p e r i o d i c w a v e e q u a t i o n a n d t h e p o t e n t i a le q u a t i o n . I n : K a c , M . ( ed .) P r o b a b i l i t y a n d R e l a t e d T o p i c s i n P h y s i c a l Sc ie nc es . L o n d o n , N e wY o r k : I n t e r s c i e n c e P u b l i s h e r s , L t d . 1 9 54

1 3. T h u r s t o n , W . : T h e g e o m e t r y a n d t o p o l o g y o f 3 - m a n i fo l d s. P r i n c e t o n U n i v e r s i t y N o t e s1 4. W a s o w , W . : T h e a c c u r a c y o f d i f fe r e n c e a p p r o x i m a t i o n s t o p l a n e D i r i c h l e t p r o b l e m s w i t hp i e c e w i se a n a l y t i c b o u n d a r y v a l u es . Q . A p p l . M a t h . l fi , 5 3 - 6 3 ( 19 5 7)

O b l a t u m 1 1 - V I - 1 9 8 6

burt rodin- schwarz's lemma for circle packings

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