OF ARBITRARY RIEMANN SURFACES

(1)

OF ARBITRARY RIEMANN SURFACES

B Y

L E O S A R I O

University of California, Los Angeles, U.S.A. (1)

1. W e consider a n a l y t i c m a p p i n g s of a n o p e n R i e m a n n surface R i n t o a closed R i e m a n n surface S.

T h e first a n d s e c o n d m a i n t h e o r e m s of t h e classical N e v a n l i n n a - A h l f o r s [3] t h e o r y were g e n e r a l i z e d in 1960 b y S. C h e r n [4] t o a n R o b t a i n e d f r o m a closed surface b y o m i t t i n g a f i n i t e n u m b e r of p o i n t s . H e also r e f e r r e d to a f o r t h c o m i n g p a p e r where, in a d d i t i o n , a f i n i t e n u m b e r of d i s k s a r e r e m o v e d .

C h e r n ' s e l e g a n t w o r k s t i m u l a t e d t h e p r e s e n t a u t h o r t o l o o k i n t o t h e q u e s t i o n : w h a t can b e s a i d a b o u t a r b i t r a r y o p e n R i e m a n n surfaces R? I n p a r t i c u l a r , c a n N e v a n l i n n a ' s first a n d s e c o n d m a i n t h e o r e m s a n d a n y c o u n t e r p a r t of P i c a r d ' s g r e a t t h e o r e m b e e s t a b l i s h e d o n t h e m ? A p r i o r i t h e r e s e e m e d to b e no basis for a c o n j e c t u r e : i t w a s k n o w n f r o m classification t h e o r y t h a t w h e n t h e g e n u s b e c o m e s infinite, i n t u i t i o n c a n no l o n g e r be r e l i e d u p o n , a n d surprises a r e possible. Moreover, a l t h o u g h J . T a m u r a [12] h a d singled o u t a class of m e r o m o r p h i c f u n c t i o n s w i t h a t m o s t t w o P i c a r d v a l u e s , M. H e i n s [5] h a d e x h i b i t e d a p a r a b o l i c R i e m a n n s u r f a c e w i t h one b o u n d a r y c o m p o - n e n t , w h i c h c a r r i e d a m e r o m o r p h i c f u n c t i o n w i t h a n i n f i n i t e n u m b e r of P i c a r d v a l u e s . This r a t h e r s e e m e d t o s p e a k a g a i n s t a n y s e c o n d m a i n t h e o r e m in t h e g e n e r a l case.

I n t h e p r e s e n t p a p e r we p r o p o s e a new choice for t h e p r o x i m i t y f u n c t i o n a n d t h e c h a r a c t e r i s t i c f u n c t i o n . T h e first m a i n t h e o r e m , t h e s e c o n d m a i n t h e o r e m , a n d t h e d e f e c t a n d r a m i f i c a t i o n r e l a t i o n s can t h e n be g i v e n for a r b i t r a r y R i e m a n n surfaces (1) An hour-address delivered at the meeting of the Mathematical Society of Japan in Tokyo on September 14, 1962.

The work was sponsored by the U.S. Army Research Office (Durham), Grant No. DA-ARO(D)- 31-124-G40, University of California, Los Angeles. The author is also grateful to his student and present colleague, Dr. K. V. R. Rao, who read the manuscript and made his valued comments.

1 - 632917. Acta mathematica. 109. Imprim6 le 28 mars 1963.

(2)

2 LEO SARIO

R (Theorems l~ 2, 3). O u r f o r m of t h e second m a i n t h e o r e m is universally valid, w i t h o u t exceptional intervals. Meromorphic functions on an a r b i t r a r y R are a special case, a n d earlier results are included (Corollaries 1-4). I n particular, for functions in t h e plane or t h e disk we h a v e n e w simple proofs for classical theorems.

I n c o n t r a s t with Chern's paper, w r i t t e n for differential geometers, our m e t h o d is elementary, t h e m o s t sophisticated tool used being Stokes' formula. T h e m e t h o d also applies, m u t a t i s m u t a n d i s , to open image surfaces S [11].

w 1. Proximity function

2. L e t R be an a r b i t r a r y open R i e m a n n surface a n d S an a r b i t r a r y closed Rie- m a n n surface. On R we use t h e s y m b o l $ b o t h for t h e generic p o i n t a n d t h e local variable. On S we similarly e m p l o y ~. A m a p p i n g ~ = ](z) into S is, b y definition, a n a l y t i c if it is so in terms of local variables.

On S consider points a 1 .. . . . ao. T h e question is, in essence, h o w m a n y a~'s can be P i c a r d points. First we need a f u n c t i o n to measure t h e p r o x i m i t y of two points o n S .

T a k e ~0, ~1 E S different f r o m a 1 .. . . . ao. C o n s t r u c t on S - ~o - ~1 t h e h a r m o n i c function t o with t0(~) + 2 log ] ~ - ~0 [ h a r m o n i c a t ~o, a n d to(~) - 2 log I ~ - ~1 [ h a r m o n i c a t

~i, t h e f o r m e r tending to 0 as ~-->~0 in a fixed p a r a m e t r i c disk. Set

s0(~) = log (1 + et~162 (1)

F o r a 4 = ~0 let t(~, a) be h a r m o n i c on S - a - ~0 with t(~, a) + 2 log I ~ - a l h a r m o n i c at a, a n d t ( $ , a ) - 2 1 o g I ~ - ~ 0 ] h a r m o n i c at ~0, t h e latter tending to So(a ) as ~-->~0.

T h e function

s(~, a) = t(~, a) + s0(~ ) (2)

becomes logarithmically infinite as ~-->a a n d is b o u n d e d below on S. I t is t h e proxi- m i t y function we set o u t to construct.

T h e function is s y m m e t r i c : for a n y a, b E S ,

s(a, b) = s(b, a). (3)

This is seen directly b y a p p l y i n g Stokes' formula to t(~, a) a n d t(~,b)along small circles a b o u t a, b, ~0 a n d t h e n letting t h e circles t e n d to these points.

3. I n terms of s we endow S with a conformal metric. T h e Euclidian area ele- m e n t dS in t h e p a r a m e t r i c ~-disk is given the area

dec = ]~ dS, (4)

(3)

where ~2 = As --

et~

[grad t o 12 (5) (1 + et~ 2

Clearly dco is a conformally i n v a r i a n t area element in this metric.

T h r o u g h o u t this paper we d e n o t e b y v(~) t h e n u m b e r of zeros of the " f u n c t i o n "

in question. W e h a v e

r(2) = es

+ 2 = 2g, (6)

where

es

is t h e Euler characteristic of

S,

a n d g is its genus. I n fact, t h e zeros of ;t are those of g r a d to, a n d this v e c t o r forms a differentiable v e c t o r field on S - t 0 - $ 1 - B y Lefschetz's fixed point t h e o r e m the n u m b e r of its singularities is t h e Euler characteristic of S - r r t h a t is,

es

+ 2. This in t u r n is 2g. T h e proof can also be based

9 $

on t h e R i e m a n n - R o c h t h e o r e m or on geometric properties of t o + ~to.

T h e total area of S is

e ~ 1 7 6 1

where fl~ is t h e level line t o = x e ( - c~, cr ).

et~

+ et0) 2

dt~ dx

= 4ze, (7)

A log ,~ = 1.

F u r t h e r m o r e )2 (8)

Indeed, t h e l o g a r i t h m of t h e n u m e r a t o r of X is harmonic, a n d t h a t of t h e d e n o m i n a t o r is t h e same function s o we s t a r t e d w i t h . '

I n passing we n o t e t h a t if S is open, t h e n we require t h a t t o has, in essence, vanishing n o r m a l derivative on t h e b o u n d a r y (cf. [7]), a n d o u r reasoning holds verbatim.

As a b y - p r o d u c t we h a v e c o n s t r u c t e d on an a r b i t r a r y R i e m a n n surface a metric with finite total area a n d c o n s t a n t Gaussian curvature.

w 2. First main theorem

4. On R choose a p a r a m e t r i c disk R 0 with b o u n d a r y flo such that/(flo) does n o t m e e t a 1 ... aq, to, r L e t ~ be an a d j a c e n t regular region with b o u n d a r y flo U fin. On f o r m t h e h a r m o n i c function u with u = 0 on fl0, u = k ( ~ ) = const, on

fin, f~o du= 1.*

F o r a n y h E [0, k] let flh be t h e level line u = h a n d d e n o t e b y ~h t h e region be- tween flo a n d fib. B y t h e same reasoning as before, t h e n u m b e r of zeros of g r a d u is t h e Euler characteristic

e(h)

of ~h:

v(h, g r a d u) =

e(h).

(9)

(4)

4 LEO SARIO

5. W e are n o w r e a d y to prove t h e first m a i n theorem. L e t a be a n y p o i n t on S a n d let zj be its inverse images in ~h. D e n o t e their n u m b e r , counted with multi- plicities, b y

v(h,a).~Let

Aj be small disks a b o u t zj with boundaries aj, oriented clockwise; flh a n d fl0 are oriented to leave R 0 to t h e left. A p p l y Stokes' f o r m u l a t o

v(z)

= h - u(z)

^{a n d}

s(/(z),

^a):

f ~.j§ ~._~o V dS* - s dv * = f a~_oAj v • s dR. (10) H e r e

dR

is t h e Euclidian area element in t h e z-disk.

As t h e aj shrink to t h e zj, t h e second t e r m in f ~ j goes to zero, for t h e flux of v across ~l vanishes. The flux of s tends to 4z~ a n d

--->4~v(zs)=4~ (h-x)d~(x,a).

I n t e g r a t e b y p a r t s a n d d e n o t e this c o n t r i b u t i o n of t h e e / s b y

A(h,a) =4e f]~(x,a) dx.

(11)

I t is our

counting [unction.

Clearly it vanishes for a P i c a r d p o i n t a.

I n f~h, v = 0 , a n d we designate this p a r t corresponding to flh b y

B(h, a) = f~hsdu.*

⁽¹²⁾

This we choose as our

proximity /unction.

I t gives t h e m e a n p r o x i m i t y to a of t h e image of fib.

On t h e right we have, in terms of

doJ(/(z)),

C(h) = f~h vdo).

⁽¹³⁾

This is our

characteristic /unction.

I t is i n d e p e n d e n t of a.

W h a t r e m a i n are t h e integrals along rio:

D(h, a) = B(O, a) + hB'(O, a). (14)

N o t e t h a t this is

O(h).

T h e o n l y functions of interest t u r n o u t to be those for which

C(h)

grows m o r e rapidly t h a n h. T h u s D is a negligible remainder. This is even more clearly seen if we do n o t o m i t R 0 f r o m ~ , in which case there are no integTals along rio. B u t for o u r later results t h e present set-up is more suitable.

(5)

We have established the first main theorem:

T H E O R E M 1.

For analytic mappings o] an arbitrary R into a closed S, and /or any ~ oR,

A(k, a) + B(]c, a) = C(]c) + D(k, a). (15) Thus the beautiful balance continues to hold despite infinite genus: a n y defect in coverage of a point a, such as of a Picard point, is compensated for b y a close p r o x i m i t y of /(fib). The functions A and B add up, in essence, to the same C for all a.

6. The characteristic C retains its simple geometric meaning. To see this take v = 1 in (10). The integrand on the left is then

ds,*

on t h e right, A,s

dR=dto(/(z)),

and we h a v e

A'(h, a) + B'(h, a) = [" do) + D'(h, a).

J l'l h

A comparison with (15) gives

C'(h) = J~h dto.

(16)

We see t h a t the characteristic is the integral of the total area of the multisheeted image of ~h over S.

7. We n e x t ask how numerous are the points a with a Pic~rd nature, i.e., with great contributions from B. Do t h e y continue to be exceptional compared with strongly covered points a? To answer this we m u s t estimate B upward.

This process is facilitated b y replacing B b y the integral of its integral. For heuristic motivation consider the simple case of the exponential function e ~. I t m a p s not only the boundaries of exhausting regions but, naturally, also these regions them- selves, with increasing mean proximity to the Picard points 0 and c~. This phen- omenon is universal, and we can replace the conventional curvilinear proximity b y areal proximity. To facilitate subsequent computations, we even integrate this.

We introduce for a n y function ~(h) the notations

q~,(h)=;cf(x)dx, cf2(h)=;cfl(x)dx.

(17)

Then the new p r o x i m i t y function is B~, and we can a t t a c h subindices

"2"

to each t e r m in (15).

8. Another simple device t h a t will shorten later reasoning is the following. We add to a I . . . . ,aq the 2g zeros aq+l . . . . ,aq+2g of 2 a n d set for a n y function

y~(h,a),

(6)

6

Then (15) gives

L E O S A R I O

q + e g

y~(h) = ~ y~(h, a,).

A.,.(h) +

B2(h ) = (q + 2g)

Ce(h ) + De(h ).

Our task is to find an upper estimate for Be(h).

(18)

(19)

w 3. Second main theorem

9. Set a(~) = exp [~+2g

s(~,

a,) - 2 log (~+2g s(~, at) + eonst.)] with ~ s +const. > 0 on S, and distribute on S the mass

dm = a deo.

I t has singularities at the as. The total mass

m = f s d m

is finite, however.

The density a2 z of

dm

induces in the u + iu*-plane the density a/~ 2, where

tu(z) = 2(z) l /' (z) [1

grad u(z)] -1. (20) We use the concavity of the logarithm to obtain

~< I_ log

adu + 2*

log

(B(h) +

eonst.) (21)

B(h)

J # h

and decompose the integral into

F(h)

= ..I% log

(a I ~e ) du,*

(22) G(h) = - J ~h

log

#e du .*

Then

B~(h) <~ F2(h ) + G~(h) +

2[log

(C(h) +

O(h))]z. (23) We shall first estimate Fe, then G 2.

10. We write

H(h) = f~h alae du *

(2 4)

and obtain by the concavity of the logarithm,

F(h) <~

log

H(h),

Fl(h) < h log (~ Hl(h)) = h log Hi(h) - h log h,

Fe(h ) < h e

log

He(h) + O(h 2

log h). (25)

(7)

To estimate H~ note t h a t

Hi(h ) = f sv(h, a) dm(a).

In fact, both quantities give the total mass on the multisheeted image of ~h over S. On integrating (15) with respect to

din(a)

the first term thus gives 4~tH=(h). Since s(~,

a)=s(a,~)

is uniformly bounded below on the compact S , - B ( h ) contributes 0(1) and we obtain

4:~ H~(h) <

mC(h) § O(h).

Substitution into (25) yields

F~(h) < h 2

log

(C(h) + O(h)).

(26)

11. To obtain

G=(h)

we first evaluate

G'(h) = - 2 ~ d*

_{log F.}

JJ ~h

Let Fj be small disks about the singularities of log /~, bounded by clockwise oriented ~r Then

f a

: 2 / d ~r log ~ - - 2 / i z log

G'(h) 2dR.

JE

y j - f l o J E~h- u F i

B y virtue of (20), (9), (4), (8), and (16), we have

G'(h)

= 4~[ -

v(h, 2) -

~(h,/') § e(h)] + 2C'(h) § const. (27) Triple integration gives

G=(h).

12. In the last term of (23) we replace the integrand in both integrations b y its value at the right end point of the interval of integration and obtain the estimate 2h 2 log

(C(h)+ O(h)).

We substitute it together with

G~(h)

and (26) into (23), this into (19), and set

f:

E(h) = 4~ e(x) dx.

We have reached our main result, the seeond main theorem:

T H E ORE ~ 2.

For an analytic mapping / o/ an arbitrary open Riemann sur/ace R into a closed Riemann sur/ace S, any regular subregion ~ c R with k = k(~) gives

q

(q + es) C2(k) < ~ A2(k, as) -

A2(k,/') + E~(k) + O(k s + k 2 log C(k)). (28) As a special case we have the second main theorem for meromorphic functions on arbitrary Riemann surfaces.

(8)

8 LEO SARIO

w 4.

Consequences

13. I n t h e sequel we consider n o n d e g e n e r a t e m a p p i n g s [ characterized b y

lira k3 + ]ca log C(k) = 0. (29)

n--,R C~(k)

T h e condition is assured b y C(k)/k-->c~ a n d t h e existence of a c o n s t a n t 0 < : r 1 with C(:r C(k)-->c~. (This f o r m u l a t i o n was suggested to the a u t h o r b y K . V . R . Rao.) The condition is general a n d is obviously even m e t b y such functions as k ~ a n d e "k, v = const. > 1, 0, respectively.

14. Using canonical regions ~ we i n t r o d u c e the defect As(k, a)

5(a) = 1 - lim , (30)

a--,R C~(k) t h e ramification index

a n d t h e Euler index

v~=lim Aa(k' 1') (31)

n . R Ca(k) '

= lim Ea(]c) (32)

h~-R C~(k)"

F r o m (28) we o b t a i n t h e following defect a n d ramification relation:

T H E O REM 3. F o r nondegenerate m a p p i n g s o/ an arbitrary R into a closed S,

Z ~(a) + ~ <~ ~ - es. (33)

I n particular there can be at m o s t ~7- es P i c a r d values.

I f S is o b t a i n e d f r o m a closed surface S o b y r e m o v i n g n points then, in terms of quantities defined for S o ,

~(a) + ~ ~ ~ - eso - n. (33')

15. We have t h e following i m m e d i a t e consequences. First consider t h e existence of m a p p i n g s /.

C O R O L L A R Y 1. A necessary c o n d i t i o n / o r a nontegenerate m a p p i n g o / a n arbitrary R into a closed S is that ~ >1 es. For a closed S less n points, ~ m u s t dominate es + n.

:For the sphere S or the plane or the p u n c t u r e d plane we have es >~ 0 a n d there is no restriction. This is compatible with t h e t h e o r e m of B e h n k e a n d Stein: e v e r y open R i e m a n n surface can be m a p p e d into the p u n c t u r e d plane. F o r t h e torus S we again h a v e no restriction b u t for genus > 1, C(]r c a n n o t grow more r a p i d l y t h a n E(k).

(9)

16. N e x t suppose R has finite Euler c h a r a c t e r i s t i c .

C O R O L L A R Y 2. I ] e ( R ) < ~ , then the n u m b e r P o/ P i c a r d v a l u e s / o r a m a p p i n g into a closed S is

P <~ - es. (34)

I n t h e case where R is o b t a i n e d from a closed surface b y r e m o v i n g a finite n u m - ber of points this is t h e Chern result. W e see t h a t a finite n u m b e r of disks or a n y closed c o n n e c t e d sets can as well be removed.

17. Suppose S is t h e sphere or t h e plane.

C o R o L L A R Y 3. F o r a m e r o m o r p h i c or entire / u n c t i o n on a n arbitrary open Rie- m a n n sur/ace,

P ~ < 2 + ~ / or P ~ < I + z ] , (35)

respectively.

T h e former b o u n d was shown to be sharp in [9, 10] a n d b y B. R o d i n in [6].

F o r ~ / = 0 we h a v e t h e T a m u r a [12] functions with P~< 2.

18. As t h e m o s t special case let R be t h e finite or infinite disk.

C o R o L L A R Y 4. F o r m e r o m o r p h i c / u n c t i o n s on [ z I < ~ ~ ~ ,

(~(ai+ v ~ ~< 2. (36)

W e can n o w t a k e u = l o g r on R a n d t h e integration with respect to h covers all of R. B y L ' H o s p i t a l ' s rule t h e subscripts " 2 " in ( 3 0 ) - ( 3 2 ) c a n be d r o p p e d a n d we h a v e a new proof for t h e conventional f o r m of t h e defect relation, simultaneously for t h e plane a n d t h e disk. The second m a i n t h e o r e m (28) is valid w i t h o u t exceptional intervals.

References

[l]. L. A~LFORS & L. SARIO, R i e m a n n sur/aces. Princeton University Press, Princeton, 382 pp.

[2]. L. AHLFORS, Zur Theorie der Uberlagerungfl~chen. Acta Math., 65 (1935), 157-194.

[3]. - - , 1Jber die Anwendung differentialgeometrischer Methoden zur Untersuehung von Uberlagerungfl~tchen. Acta Soc. Sci. Fenn. Nova Ser. A . I I , 6 (1937), 17 pp.

[4]. S. C~E~N, Complex analytic mappings of Riemann surfaces I. A m e r . J . Math., 82, 2 (1960), 323-337.

[5]. M. HEINS, Riemann surfaces of infinite genus. A n n . o/ Math., 55, 2 (1952), 296-317.

[6]. B. RODIN, Reproducing ]ormulas on R i e m a n n sur]aces. Doctoral dissertation, Univer- sity of California, Los Angeles, 1961, 71 pp.

[7]. L. SARIO, A linear operator method on arbitrary Riemann surfaces. Trans. A m e r . 21lath.

Soc., 72 (1952), 281-295.

(10)

10 LEO SARIO

[8]. - - - , P i c a r d ' s g r e a t t h e o r e m on R i e m a n n surfaces. Amer. Math. Monthly~ 69 (1962), 598-608.

[ 9 ] . - - - , M e r o m o r p h i c f u n c t i o n s a n d c o n f o r m a l m e t r i c s on R i e m a n n surfaces. Pacific J . Math., 12 (1962), 1079-1098.

[10]. - - - , Islands a n d pensinsulas on a r b i t r a r y R i e m a n n surfaces. Trans. Amer. Math. Soc., 106 (1963), 521-532.

[11]. - - , A n a l y t i c m a p p i n g s b e t w e e n a r b i t r a r y R i e m a n n surfaces ( R e s e a r c h a n n o u n c e m e n t ) . Bull. Amer. Math. Soc. 68 (1962), 633-638.

[12]. J . TAMURA, M e r o m o r p h i c f u n c t i o n s on o p e n R i e m a n n surfaces. Sci. Papers Coll. Gen.

Ed. Univ. Tokyo, 9, 2 (1959), 175-186.

Received August 6, 1962

OF ARBITRARY RIEMANN SURFACES