CURVE FAMILIES F* LOCALLY THE LEVEL CURVES OF A PSEUDOHARMONIC FUNCTION
B Y
JAMES J E N K I N S and MARSTON MORSE
Johns Hopkins Institute for
University Advanced Study
Introduction
The family F * m a y be defined over an arbitrary open Riemann surface Q. When Q is not simply connected there m a y exist no single-valued P H [pseudoharmonic] function on Q with F * as its family of level lines. On the universal covering surface M of Q there do exist P H functions u, single-valued on M and with a family F ~ of level lines which projects into F * on Q. While u m a y not be single-valued on Q it m a y behave like an integral in that it has branches which differ by a constant, or it m a y have a real logarithm which has this property. I n studying such behavior of u one m a y focus on the branches of u obtained by continuation of u along a single closed curve k not homotopic to zero on Q.
I n this way one is led to the essentially typical case of a family F * defined on a sphere Z* with a north pole N and south pole S removed. Although there m a y be no single-valued P H function u on Z* with F * as its family of level lines there will in general be multiple- valued functions u satisfying linear relations
(1.0) u [ p (1)] = a u ( p ) + b (a I= O)
where p and p(1) are points on the universal covering surface M of Y~*, and where p and p(ll in M project into the same point in Z*, but on M have longitudes 0 and 0 + 2 Jr respec- tively. However the values of the constants a and b for which a relation (I.0) m a y hold depend in a deep way upon the nature of the family F*. See MJ 4 and MJ 5.
I n the present paper we decompose ~:* into canonical regions, "primitives," "caps,"
"annuli," "polar sectors," " c u t sectors," etc., whose nature is determined by F*. With F we associate integral indices v (F) and # (F) [defined in a later paper]. The existence of P H functions u satisfying prescribed linear relations (1.0) depends upon these indices and upon the character of the decomposition of Y~*.
1 --533807. Acta Mathematica. 91. I m p r i m 6 lo 18 m a i 1954.
J A M E S J E N K I N S A N D M A R S T O N I ~ O R S E
W h e n F * is n o n - s i n g u l a r K a p l a n [3] h a s g i v e n a d e c o m p o s i t i o n of Y~* in s o m e b u t n o t in all of t h e cases we f i n d e s s e n t i a l . K a p l a n ' s r e s u l t s are less g e n e r a l in a n a priori sense t h a n ours e v e n in t h e n o n - s i n g u l a r case in t h a t he r e q u i r e s t h e c u r v e s in F * t o be h o m e o m o r p h i c m a p p i n g s of i n t e r v a l s a n d circles. W e c o n f i r m m a n y of K a p l a n ' s t h e o r e m s p a r t i c u l a r l y o n a s y m p t o t e s , a n d a d d t h e r e s u l t s n e c e s s a r y for o u r p u r p o s e s . O u r m a i n t h e o r e m s a r e s t r o n g l y a f f e c t e d b y t h e p r e s e n c e of s i n g u l a r p o i n t s in F * . See also B o o t h b y [1, 2] a n d T S k i [5].
w 2. R e v i e w a n d e x t e n s i o n s
L e t E b e a 2-sphere. L e t N a n d S be d i a m e t r i c a l l y o p p o s i t e p o l e s in E, t e r m e d re- s p e c t i v e l y t h e n o r t h a n d s o u t h poles of Z. S e t
Z * = Z - N - S .
L e t ~o be a d i s c r e t e set of p o i n t s in Y~* a n d set G = ~]* - w . C o n s i d e r a f a m i l y F of o p e n a r c s or t o p [topological] circles in G s u p p o s i n g t h a t F c o n t a i n s a u n i q u e e l e m e n t :r m e e t i n g a n y g i v e n p o i n t pEG. A n open arc ill G is u n d e r s t o o d as tile i m a g e in a 1 - 1 c o n t i n u o u s m a p p i n g i n t o G of a n o p e n i n t e r v a l . T h e a r c is t h e i m a g e a n d n o t t h e m a p p i n g . A t o p circle in G is t i l e h o m e o m o r p h in G of a circle.
F-neighborhoods Xv. L e t n bc t h e o p e n disc { Iw] < 1} in t h e eomI)lcx w-plane. W i t h each p o i n t pE)-]* t h e r e shall be a s s o c i a t e d a n F - n e i g h b o r h o o d X v of p w i t h )Cv c G U p, a n d a t o p m a p p i n g of )~v o n t o 1) which sen(ls p i n t o w -- 0 a n d t h e m a x i m a l o p e n arcs of F I X ~ i n t o t h e m a x i m a l o p e n level a r c s of ~ w n, in D,n > 0, [with w = 0 e x c l u d e d w h e n n > 1]. W e s u p p o s e n = 1 for pEG a n d n > 1 for peso. P o i n t s in eo a r c t e r m e d singular p o i n t s of F . T h e v a l u e wED is t e r m e d a c a n o n i c a l p a r a m e t e r of its a n t e c e d e n t in Xv.
T h e o p e n a r c s of F [ ( X v - p) i n c i d e n t w i t h p are t e r m e d F-rays of Xv. T h e s e r a y s d i v i d e X v - p i n t o 2 n o p e n regions t e r m e d F-sectors of X v i n c i d e n t w i t h p.
Right N. W i t h each pEG we also a s s o c i a t e a n e i g h b o r h o o d N v of p in G a n d a h o m e o - m o r p h i c m a p p i n g of _~v o n t o a s q u a r e K : ( - 1 _<u < 1 ) ( - 1 ~ v ~ 1) such t h a t p goes i n t o t h e origin in K a n d t h e m a x i m a l s u b a r c s of FI~V v go i n t o a r c s u = c , - 1 ~ v g 1, where e is a c o n s t a n t in t h e i n t e r v a l [ - 1,1]. W e refer t o N v as a r i g h t n e i g h b o r h o o d or r i g h t N v of p a n d t e r m (u,v) canonical coordinates of t h e a n t e c e d e n t in N r of (u,v) in K . A n o p e n arc /t in N v on which v = T ( u ) ( - 1 < u < 1), where q~ is s i n g l e - v a l u e d a n d c o n t i n u o u s , is called a t r a n s v e r s a l of Nv. T h e o p e n arc in N v on w h i c h v = 0, - 1 < u < 1, is called t h e principal transversal of Nv. More g e n e r a l l y a t r a n s v e r s a l / x shall be a n y o p e n a r c in G e a c h p o i n t of w h i c h is in a n o p e n s u b a r c of ~t which is a t r a n s v e r s a l of some r i g h t N~. A t r a n s v e r s a l w i t h a closure in Y~ which is a n a r c in G is t h e p r i n c i p a l t r a n s -
C U R V E F A M I L I E S F * L O C A L L Y T H E L E V E L C U R V E S O F A P S E U D O I t A R M O N I C F U N C T I O N 3
versal of a suitably chosen r i g h t Nv. A non-singular arc in Z * is t e r m e d a transverse arc if a subarc of some transversal. I f Y is an F - s e c t o r incident with p, a transversal meeting each element of F l Y a n d with p as limiting end point is called a transversal ray of Y in- cident with p. Transversal r a y s ~t a n d # incident with p, b u t in different F-sectors of X v incident with p, define a transversal cut ~p~ of X~.
F-vectors. A n y sensed s u b a r c of an :rE F will be called an F - v e c t o r . B y definition a n F - v e c t o r is simple a n d closed in G, a n d never a t o p circle. E a c h F - v e c t o r is in some r i g h t N~ [MJ 2 w 3].
Coherent sensing. L e t each r162 F be given a sense. The resulting f a m i l y F s of sensed will be called a sensed image of F . W e shall refer to a continuous d e f o r m a t i o n A of an F - v e c t o r A in the space of F - v e c t o r s with a F r d c h e t metric. The sense of an image of A u n d e r A shall be d e t e r m i n e d b y t h e A - i m a g e s of t h e initial a n d final points of A. W e s a y t h a t F s is c o h e r e n t l y sensed if a n y c o n t i n u o u s d e f o r m a t i o n A of an F - v e c t o r A (initially sensed as in F 8) t h r o u g h F - v e c t o r s sensed b y ~ is necessarily t h r o u g h F - v e c t o r s sensed b y F s.
W e say t h a t F is coherent if it a d m i t s a c o h e r e n t l y sensed image F s, otherwise non-coherent.
A family F on Z* m a y be c o h e r e n t or n o n - c o h e r e n t as t h e following examples show.
Examples. L e t Z * be represented b y the z-plane, with z =~ 0, S b y t h e origin, a n d N b y t h e p o i n t a t iafinity. I f z = x + iy, t h e level lines of x on Z * afford a coherent family. T h e loci on which
X 2
y = ~ - a - a ( a > 0 )
t a k e n with the open arc on which x = O, y > 0 afford a n o n - c o h e r e n t family.
W e shall establish the following theorem.
Theorem 2.1. A necessary and su]/icient condition that F be coherent over G is that, taken over some neighborhood o / N or o / S , F admit two distinct coherent sensings.
This follows as in t h e proof of T h 4.2 of M J 2.
T h e / a m i l y F*. T h e f a m i l y F * shall consist of elements h, k, m .... in E * which are t o p circles or open arcs in E*. I f a non-singular point p is in h, h shall contain t h e open arc a v e F meeting p a n d a n y limiting end point or points of a v E E * . A n hE F * comprising just one ~E F is called non-singular.
Positive and negative limit points. L e t an open arc h E F * be sensed a n d be given a
1 - 1 representation in which p (t) is t h e 1 - 1 continuous image of t, - co < t < o% with
t increasing in the positive sense of h. B y a positive (negative) limit point of h is m e a n t a n y p o i n t in E which is a limit p o i n t of a sequence of points p (t~), where t , increases (de- creases) w i t h o u t limit as n ~ oo.
4 J A M E S J E N K I N S AND MARSTON MORSE
Covering sur/aces M and /amilies F M. L e t K be a n y open, o r i e n t e d a n d c o n n e c t e d R i e m a n n surface a n d w, F , "F* a d m i s s i b l y d e f i n e d for K as a b o v e for Z * . L e t M be a r e l a t i v e l y u n b r a n c h e d , u n b o r d e r e d c o v e r i n g surface of K . L e t aM be a n y m a x i m a l o p e n a r c or t o p circle in M " c o v e r i n g " a n e l e m e n t ~ E F . I f cr is a n o p e n arc, ~ M is a n o p e n arc.
I f ~ is a t o p eircle, 0~ M is a n o p e n a r c or a i l o m e o m o r p h i c t o p circle. T h e t o t a l i t y of t h e e l e m e n t s 0~ M c o v e r i n g t h e e l e m e n t s gE F f o r m s a f a m i l y FM in M . L e t O) M be t h e set of p o i n t s in M covering t i l e set ~o in 22". T h e f a m i l y FM i n c l u d e s one a n d o n l y one e l e m e n t m e e t i n g each p o i n t of M - o ~ . J u s t as F * was d e f i n e d in Z * w i t h t h e a i d of F a n d ~o, so h e r e F ~ is d e f i n c d in M in t e r m s of FM a n d e0M.
I n the remainder o/ this paper M shall be the universal covering sur/ace o/22".
F r o m M J 2 we infer t h e following.
T h e o r e m 2.2. Let the universal covering sur/ace M o/ "Z* be considered as a top sphere H /rom which one point Z has been removed.
(1) Then any open arc O~ME FM has limiting end points in H, distinct unless coincident with Z. Each end point o/ O~ M di[/erent /tom Z is a point o/WM. Cf. M J 2 T h 7.1.
(2) There are no top circles in F~.
(3) For any open arc hME F ~ the positive and negative limit sets reduce to Z. Cf. M J 2 Th 7.2.
Corollary 2.1. No two top circles in F* can intersect or be joined by a subarc o / a n element in F*.
I f two t o p circles gl a n d g2 ill F * m e t t h e r e w o u l d be a f i n i t e sequence of e l e m e n t s of F in gl U g2 whose closure w o u l d c a r r y a closed c u r v e g (not n e c e s s a r i l y s i m p l e ) b o u n d i n g a r e g i o n in 22*. A s u i t a b l y chosen closed c u r v e gM c o v e r i n g g wouhl be s i m p l e w i t h c a r r i e r in F ~ , c o n t r a r y to T h 2.2(2). T h e s e c o n d a f f i r m a t i o n of t h e c o r o l l a r y is s i m i l a r l y e s t a b l i s h e d .
F *
T h e o r e m 2.3. I[ M is the universal covering sur[ace o[ 22* no hE M intersects a trans- versal or a transversal cut in M in more than one point.
Since M is t h e h o m e o m o r p h of a f i n i t e z - p l a n e this follows f r o m Cor. 7.5 of M J 2.
w 3. F-sets and F-regions
W i t h F t h e r e a r e n a t u r a l l y a s s o c i a t e d c e r t a i n sets a n d regions w h i c h we n o w define.
B y a region we shall a l w a y s m e a n a n open, c o n n e c t e d set.
F.sets. A set H c Z will be t e r m e d a n F - s e t , if w h e n e v e r a n o n - s i n g u l a r p o i n t p is in H , t h e a r E F m e e t i n g p is also in H . F r o m t h e n a t u r e of a r i g h t Np of p i t is clear t h a t t h e
CURVE FAMILIES F * LOCALLY THE LEVEL CURVES OF A PSEUDOHARMONIC FUNCTIOI~ 5
complement, closure or b o u n d a r y of an F - s e t is an F-set. T h e intersection, or union of a n y ensemble of F - s e t s is also an F - s e t .
Conditions O. A n open set R in Z which is an F - s e t will be said to satisfy Conditions 0 if t h e inclusion of a p o i n t p of Z * in fl R implies t h a t a n y sector of a sufficiently restricted F - n e i g h b o r h o o d X~ of p is in R or in Z - / ~ , a n d at least t w o of the F - r a y s b o u n d i n g sectors of X v N R are in fl R.
F-regions. A s i m p l y connected region R in Z whose b o u n d a r y consists of more t h a n one point a n d which satisfies Conditions O will be called an F-region.
The boundary fiR. I f R is an F - r e g i o n f i R is an F-set. L e t p E f l R be non-singular a n d suppose t h a t ~v E F meets p. A n y sufficiently small right n e i g h b o r h o o d N v of p is separated b y :r into t w o regions of which at least one a n d possibly b o t h are in R.
U p to this point we h a v e n o t used curves (which are mappings) b u t r a t h e r sets such as open arcs a n d t o p circles. We n o w i n t r o d u c e open curves a n d closed curves as continuous m a p p i n g s into Z of sensed open arcs a n d circles respectively. T w o such curves ql a n d ~2 are the same or more precisely in t h e same curve class if ql = ~2 T, where T is a t o p sense preserving m a p p i n g of t h e d o m a i n of T1 onto the d o m a i n of ~2. I f T is a t o p m a p p i n g of t h e d o m a i n of a curve ~ o n t o itself inverting sense, t h e n ~ T will be d e n o t e d b y ~ - . W e m a y denote ~ b y ~ + .
If ~0 is a m a p p i n g of a d o m a i n E into Z defining a curve, the image ~ ( E ) in Z will be t e r m e d t h e carrier ]~01 of q. B y t h e intersection of two curves q a n d v 2 we m e a n the inter- section of [~1 and ]91. B y definition a curve ~ b o u n d s a set E if f i E = ]q[.
R-continuations in fl R. Suppose Z oriented so t h a t t h e local r i g h t (left) sets of a n y point in an open sensed arc are well defined, cf. M J 1 w 5. L e t R be a n F - r e g i o n a n d a E F be in fl R. L e t cr be sensed so t h a t its local right sets are in R. T h e n R can be c o n t i n u e d as a locally simple curve q, cf. Morse [4], so t h a t its carrier is an F-set, a n d so t h a t the sensed carrier of a simple open s u b c u r v e of ~ has its local right sets in R. C o n t i n u e d m a x i m a l l y in this w a y with carrier in F *
If R,
~ will be called a right R - c o n t i n u a t i o n in fiR. Le/t R- c o n t i n u a t i o n s in f i R are similarly defined. T w o R - c o n t i n u a t i o n s ql a n d ~2 are regarded as the same if a n d only if t h e y are b o t h right or b o t h left continuations, a n d if ~1 a n d T2 are in t h e same curve class.We need a p a r a m e t e r i z a t i o n of t h e b o u n d a r y of an F-region. W e shall m a k e use of an open disc D { I w I < 1 } a n d suppose t h a t fl D is given t h e counter-clockwise sense in t h e w-plane, so t h a t local left sets of fl D are in D.
Theorem 3.1. An F-region R is the 1 - 1 image in a directly con/ormal map ] o~ an open disc D ( Iwl < 1} onto R. A n y such map admits a continuous extension over D such that f l D
J A M E S J E N K I N S A N D M A R S T O N M O R S E
is mapped onto fl R. This mapping is 1 - 1 in a su//iciently small neighborhood relative to D o / a n y point o/fl D whose image is in Z*. The antecedent in fl D o / N or o / S is a nowhere dense and possibly empty set. The mapping
(3.1) q~ = / [ r i D
de/ines a closed curve bounding R o/which ~ R is the carrier.
One first defines / o v e r D. B y well k n o w n t h e o r e m s one can e x t e n d / as s t a t e d o v e r a set D 1 such t h a t / ( D 1 ) c o v e r s / ~ N Z * . I f f i r does n o t i n c l u d e N or S t h e p r o o f is c o m p l e t e . To c o n t i n u e c o n s i d e r t h e case in which fl R i n c l u d e s b o t h N a n d S.
L e t DN a n d Ds be r e s p e c t i v e l y t h e / - a n t e c e d e n t s of t h e i n t e r s e c t i o n of R N •* w i t h t h e n o r t h e r n a n d s o u t h e r n h e m i s p h e r e s of Z. L e t E be t h e set of p o i n t s in flDN a t w h i c h / is n o t y e t d e f i n e d . S e t / ( z ) = N for zEE. W e s h a l l s h o w t h a t / a s e x t e n d e d is c o n t i n u o u s a t each p o i n t z 0 of E .
N o w E is b o u n d e d f r o m Ds so t h a t / is c o n t i n u o u s a t z 0 i f / [ DN is c o n t i n u o u s a t z 0.
L e t zn, n = 1, 2 . . . . t h e n be a n a r b i t r a r y sequence of p o i n t s in DN t e n d i n g t o z 0. T h e n /(zn)--> N . O t h e r w i s e t h e set of p o i n t s /(zn) w o u l d h a v e a n a c c u m u l a t i o n p o i n t p in / ( / ) N ) - N a t which / - l ( p ) is well d e f i n e d a n d h a s a f i n i t e set of v a l u e s a 1 .. . . . am, n o t in E a n d in n u m b c r a t m o s t t h e n u m b e r of d i f f e r e n t F - s e c t o r s in a n F - n e i g h b o r h o o d of p. I n a suf- f i c i e n t l y s m a l l n e i g h b o r h o o d of e a c h a i in D , / is well d e f i n e d a n d z b o u n d e d f r o m z o. F r o m t h i s c o n t r a d i c t i o n we infer t h a t / ( z ~ ) - - > N a n d t h a t / is c o n t i n u o u s a t z 0.
W e s i m i l a r l y e x t e n d / o v e r D s . T h e case in which f i r i n c l u d e s N o r S a l o n e is s i m i l a r . I f t h e a n t e c e d e n t of N in r D were dense in r i D , / ( z ) w o u l d e q u a l N o v e r s o m e a r c of fl D a n d hence be c o n s t a n t .
T h e t h e o r e m follows.
Inner cycles. A J o r d a n c u r v e ~ whose c a r r i e r is a t o p circle in F * will be c a l l e d a n i n n e r cycle.
N-loops, S-loops, NS-curves, SN-curves. L e t g be a n o p e n a r c in F * . L e t ~ b e a s i m p l e sensed c u r v e whose c a r r i e r is g. S u p p o s e t h a t g h a s a u n i q u e n e g a t i v e l i m i t p o i n t A a n d a u n i q u e p o s i t i v e l i m i t p o i n t B. T h e n e i t h e r A = B = N , or A = B = S , or A = N a n d B = S, or A = S a n d B = N. [Th 2.2(3).] T h e n ~ is called r e s p e c t i v e l y a n N - l o o p , S - l o o p , N S - c u r v e , S N - c u r v e . B y t h e exterior E9~ of a n N - l o o p [S-loop/ w i t h c a r r i e r g is m e a n t t h a t r e g i o n in E which is b o u n d e d b y ~ a n d c o n t a i n s S [N]. T h e i n t e r i o r I ~ of ~ is d e f i n e d as Z - C 1 E ~ . W e call a t t e n t i o n t o t h e f a c t t h a t an~'N- o r S - l o o p , N S - o r S N - c u r v e is in Z * . Meridians. T h e c a r r i e r g of a n N S . o r S N - c u r v e will be c a l l e d a m e r i d i a n . I f M is t h e u n i v e r s a l covering surface of ]E* one sees t h a t a gM c o v e r i n g g d i v i d e s M i n t o t w o d i s j o i n t regions.
CURVE FAMILIES F * LOCALLY THE LEVEL CURVES OF A PSEUDOHARMONIC FUNCTION 7 L e m m a 3.0. Let cf be an inner cycle or an N - or S-loop in F * , and let q be a point not in q~ if q; is an inner cycle, and in EqJ if q) is a loop. Then a transversal ray 2 incident with q meets q) in at most one point.
L e t H be either one of t h e two regions b o u n d e d b y qo if ~ is an inner cycle, a n d I q if
~o is a loop. If ~ enters H at a p o i n t p of T it leaves H at no other p o i n t r. Otherwise the subarcs of 2 a n d [ ~ ] with end points p a n d r would b o u n d a s i m p l y connected region in E containing neither N n o r S. This is impossible b y Th 2.3.
L e m m a 3.1. A n y set of N-loops (S-loops) with disjoint interiors and diameters exceeding some positive constant is finite.
I f t h e l e m m a were false for N-loops there would exist a set of N-loops ~vn, n = 1, 2 . . . . with disjoint interiors I~% a n d points p . E q , such t h a t p~--->qEZ* as n t oo. T h e n u m b e r of ~% which m e e t q is at m o s t the n u m b e r of F-sectors in an F - n e i g h b o r h o o d of q. We c a n accordingly suppose t h a t no ~% meets q. N o t e t h a t q is in no I~%. We can f u r t h e r suppose the Pn chosen in ~% so as to be in a transversal r a y ;t incident with q. Since q is in ET~
for each n, it follows f r o m L 3.0 t h a t q= meets ), only in p~. H e n c e I~% includes the open arc 2~ of )t s e p a r a t e d f r o m q b y Pn" T h u s
Iq) n N Iq~ m D ~ N ;tm.
This contradiction to the choice of the qn implies t h e lemma.
To a d e q u a t e l y describe t h e b o u n d a r y of an F - r e g i o n we m u s t define N- a n d S-circuits.
N-circuits, S-circuits. N-circuits are defined as locally simple open curves T in Y~*
whose carriers are F - s e t s which h a v e end points a t N (i.e., positive a n d negative limit sets in N) a n d which intersect themselves w i t h o u t crossing. F r o m this definition of an N - c i r c u i t t h e reader can derive the following d e c o m p o s i t i o n of an N-circuit.
W h e n ~ is an N-circuit, [(P[ carries three open arcs, a, b, c, whose closures are simple F-sets; of which ~ has t h e initial point N a n d a terminal p o i n t PEY~*, $ has the initial p o i n t P a n d terminal point P , ~ has t h e initial point P a n d terminal point N. These arcs a n d end points N a n d P derive the order
N a P b P c N
from cp. T h e t o p circle $ separates N from S. Finally a N b = c N b = 0, whiie a N c is the e m p t y set, or a half open arc with limiting end point P , or a = c.
The N - c i r c u i t ~0 carries t h e t o p circle $. L e t ~p be an inner cycle with carrier l; a n d with a sense derived f r o m ~. W e t e r m ~ t h e inner cycle of the N.circuit q~. I t is u n i q u e l y d e t e r m i n e d b y ~0. [Cot 2.1.]
J A M E S J E N K I N S A N D M A R S T O N M O R S E
W h e n ~ is a n N - c i r c u i t the interior Iq) of ~ shall be t h e region whose b o u n d a r y is I ~ I U N. This region is unique a n d does n o t Contain S.
T h e definition a n d d e c o m p o s i t i o n of a n S-circuit are similar.
The sensed closed curve q) = f [fl R o / T h 3.1. W h e n fl R m e e t s N or S let d be a n y m a x i m a l open arc in f l D such t h a t ~(d) is in Z * . T h e n ~ I d is a locally simple, open, sensed curve ~0 with end p o i n t s in N U 5, a n d l yJI is an F - s e t . B y v i r t u e of t h e c o n f o r m a l i t y of f l D, the c o n t i n u i t y o f ' / I / ) , a n d the locally simple c h a r a c t e r of ~[d, ~1 d is a left R - c o n t i n u a t i o n in fl R. W h e n fl R m e e t s n e i t h e r N nor S we set d = fl D a n d n o t e t h a t ~ = ~ I d, so t h a t is a left R - c o n t i n u a t i o n in f i R with carrier fiR. I n each of the a b o v e cases we t e r m ~ l d a m a x i m a l subcurve of q9 in Z * .
T h e following t h e o r e m is basic.
T h e o r e m 3.2. I f the boundary curve o / a n F-region R is given by (3.1) in T h 3.1 then a n y
" subcurve" y~ of q9 m a x i m a l in F~ * is a left R-continuation in fl R.
(1) Each m a x i m a l subcurve ~o of q) is either (a) an N - or S-loop, (b) an N-circuit, (c) an S-circuit, (d) an N S . or SN-curve, or (e) an inner cycle.
(2) Different types o/subcurves (b) to (c) cannot coexist. There is at most one vd o / t y p e s (b), (c) or (e). When an inner cycle q~ occurs IqJ[ = f i R .
(3) T y p e (d) occurs if and only i / f i R D N 0 S. The subcurves y~ then include lust one NS-curve Y)l, and just one 5N-curve Y)2. I f Iv21] :v l yj2[, Iv211 N 1~21 is either empty, a point, an arc, or a half open arc with one limiting end point at N or at 5.
(4) The carriers o / n o two v 2 intersect at most excepting v21 and YJ2 in (3).
L e t (q0} be the set of s u b c u r v e s y~ of ~ m a x i m a l in Z * . T h e first s t a t e m e n t in the t h e o r e m has been covered.
Proof o / ( 1 ), Caze I. T h e s u b c u r v e ~0 joins the two poles. T h e n ~0 m u s t be simple. Other- wise l y~l would c a r r y a t o p circle s e p a r a t i n g N f r o m S on R. This is impossible w h e n joins pole to pole a n d R is connected. I n Case I, ~ is a n N S - c u r v e or 5 N - c u r v e .
Case I I . Not Case I. I f ~ is simple it is clearly a n N - or S-loop, or an inner cycle. I f n o t simple ~ is locally simple, h a s a n F - s e t carrier, intersects itself w i t h o u t crossing itself a n d joins a pole Z t o Z. These are the characteristics defining a n N - or S-circuit.
Proof of (2), Case I. v 2 is a n inner cycle. I n this case fl R = I~], a n d all t y p e s o t h e r t h a n (e) are excluded.
Case I I . ~ is a n N- or 5-circuit. An inner cycle is excluded as j u s t seen. Since ]Y~I separates S f r o m N , a n N - c i r c u i t excludes a n S-circuit a n d vice versa, a n d a n y circuit excludes a n N 5 - or S N - c u r v e .
CURVE FAMILIES F * LOCALLY THE LEVEL CURVES OF A PSEUDOHARMONIC F U N C T I O ~ 9
Case I I I . yJ is a n NS- or S N - c u r v e . T h e e x c l u s i o n of t y p e s (b), (c), (e), h a s a l r e a d y b e e n e s t a b l i s h e d .
Proo/ o/ (3). I t is i m m e d i a t e t h a t ~ is of t y p e (d), o n l y if f i r i n c l u d e s N U S. C o n v e r s e l y if T is a closed curve a n d m e e t s N a n d S t h e r e is a t l e a s t one N S - c u r v e Y~I, a n d one S N - e u r v e v~2 in {~}. These curves ~01 a n d ~2 i n t e r s e c t , if a t all, as s t a t e d in Wh 3.2(3). F o r I~ll U [~21 c a n n o t c a r r y a t o p circle g; since g w o u l d e i t h e r s e p a r a t e N f r o m S, which is i m p o s s i b l e w h e n fiR ~ N U S, or b o u n d a r e g i o n n o t c o n t a i n i n g N or S, which is i m p o s s i b l e b y T h 2.2.
F i n a l l y YJ1 a n d F2 a r e u n i q u e as NS- a n d S N - c u r v e s in {~}. To see t h i s l e t RM c o v e r R on M , a n d in fl RM let ~fl M a n d ~f2M cover ~Pl a n d YJ2 r e s p e c t i v e l y . T h e r e can be no m e r i d i a n g in fiR which is n o t c o v e r e d on M b y yJ1M or b y yJ2M; o t h e r w i s e a c o v e r i n g gM of g in
RM
w o u l d d i v i d e RM. T h i s is i m p o s s i b l e since RM is a h o m e o m o r p h of R. W e i n f e r t h a t vd~
a n d y~ a r e u n i q u e NS- a n d S N - c u r v e s in {~}. W e n o t e t h a t y~l M N ~ M = 0.
Proo] o/ (4). L e t ~ be a n y e l e m e n t of {~} of t y p e (a), (b), (c), or (e). T h e n t h e closure of l~ I s e p a r a t e s E i n t o a t l e a s t two o p e n sets of which one, s a y ~2, c o n t a i n s R. A n y c l e m e n t of {~} is in R c $2. L c t p be a p o i n t of ~]. A n e i g h b o r h o o d 1t v of p r e l a t i v e to .(2 m a y be o b t a i n e d as a u n i o n of a f i n i t e n u m b e r of left sets of ~ a s s o c i a t e d w i t h p. Since ~(,) D R, H v c o n t a i n s a n e i g h b o r h o o d of p r e l a t i v e t o R. B u t if s u f f i c i e n t l y r e s t r i c t e d , t h e a b o v e left sets of ~ a r e in R, since ~ is a n R - c o n t i n u a t i o n , so t h a t H~, if s u f f i c i e n t l y r c s t r i c t c d , is a n e i g h b o r h o o d of p r e l a t i v e to R. W e s u p p o s e H v so restrictc(t.
L e t F be a n y e l e m e n t of {q~}. T h c n
I,pl nsT, c - R) ntis= nfiv.
I f ~v m e e t s ~ i t m u s t t h e n be c a r r i e d b y IvTI. B u t
I~tl
carries n o e l e m e n t in {~v} o t h e r t h a n~. T h u s i n t e r s e c t i o n s of e l e m e n t s of {7} can o c c u r o n l y for t w o e l e m e n t s of t y p e ((1). T h i s e s t a b l i s h e s T h 3.2 (4).
W e shall now give c e r t a i n d e f i n i t i o n s a n d l e m m a s useful in t h e a p p l i c a t i o n of Th 3.2.
F*-cycles. L e t R be a n y s i m p l y c o n n e c t e d r e g i o n in ~ which c o n t a i n s S a n d whose b o u n d a r y fiR is t h e u n i o n of N a n d of a f i n i t e or c o u n t a b l y i n f i n i t e set of d i s j o i n t N-h)ops.
T h e n R is a n F - r e g i o n whose b o u n d a r y b e c o m e s t h e c a r r i e r of a closed curve d e f i n e d b y t h e m a p p i n g ~ =[]riD of flD o n t o f i r as in T h 3.1. W e t e r m t h e c u r v e ~ or ~ v - a n N - cycle. S - c y c l e s are s i m i l a r l y defined.
A n N-cycle, S-cycle, o r inner cycle will be called a n F*-cycle.
Concavity. L e t R be a r e g i o n b o u n d e d in p a r t b y F * - c y c l e s , circuits, or o p e n arcs iu F * . A n y such e l e m e n t ~ will be t e r m e d c o n c a v e t o w a r d R if no e l e m e n t of F * I R has a l i m i t i n g e n d p o i n t in q. W e t e r m R c o n c a v e if no e l e m e n t in F * I R h a s a l i m i t i n g e n d p o i n t in fiR. I f a n F * - c y e l e q b o u n d s a r e g i o n H a n d is c o n c a v e t o w a r d H , ~v will be t e r m e d con-
1 0 J A M E S J E N K I N S A N D MARSTON MORSE
c a v e t o w a r d a n y set in H . I n p a r t i c u l a r a n i n n e r cycle will be t e r m e d N - c o n c a v e [ S - c o n c a v e ] if c o n c a v e t o w a r d N IS]. A n N - l o o p 9 will be t e r m e d S - c o n c a v e [ N - c o n c a v e ] if c o n c a v e t o w a r d s E 9 [19]. A n S - l o o p 9 is t e r m e d N - c o n c a v e [ S - c o n c a v e ] if c o n c a v e t o w a r d s E 9 [ I 9 ] . F*-sets. A set in X will be c a l l e d a n F * - s e t if i t c o n t a i n s e a c h h E F * w h i c h i t m e e t s . I n p a r t i c u l a r a n F * - s e t which c o n t a i n s a n ccEF c o n t a i n s a n y l i m i t i n g end p o i n t w h i c h m a y h a v e in X*. The u n i o n or i n t e r s e c t i o n of a n e n s e m b l e of F * - s e t s is a n F * - s e t . R e g i o n s b o u n d e d b y F * - e y c l e s whose m a x i m a l s u b c u r v e s in 2~* are n o n - s i n g u l a r a r e F * - s e t s . T h e following is p a r t i c u l a r l y n o t e d .
(A). I / a region R is an F*-set and i/ Y is an F-sector in R incident with a point q in fl R, then fl R includes the two F-rays in fl Y incident with q.
T h e following l e m m a is a m a j o r t o o l in a p p l y i n g T h 3.2.
L e m m a 3.2. Let W be any ensemble o/ regions R each o / w h i c h is the interior o/ a non- singular N- or S-loop, or a region bounded by a non-singular inner cycle, and set U = U n i o n
R I ( R E W ) .
(a). Then U is an F*-set.
(b). U satis]ies Conditions O.
(c). I] the regions o/ W /orm a sequence H 1 c H 2 ~ .... i] ~ * - U ~-0, and i / f H ~ i s a top circle in F / o r each n, then U is concave and f U I Z * i s simple.
W e d i s c a r d t h e t r i v i a l case in which fl U N Z * = O.
P r o o / o / (a). Since e a c h RE W is a n F * - s e t , U is a n F * - s e t .
Proo] o] (b). L e t q hc a p o i n t of/~ U in ~ * a n d let Y be a n a r h i t r a r y F - s e c t o r imd(lcnt w i t h q. W e p r o v e t h e folh)wing.
(m). I / f l ( Y 0 U) meets q then Y ~ U.
Case I. S o m e H E W c o n t a i n s a n F - s e c t o r in Y i n c i d e n t w i t h q. T h c closure of a t r a n s - v e r s a l r a y # in Y, i n c i d e n t with q, m e e t s f H in q. Since f H is a n o n - s i n g u l a r i n n e r cycle or N - or S - l o o p it c a n n o t m e e t [t o t h e r t h a n in q. [Th 2.3.] H e n c e Y is in H a n d so in U.
Case I I . N o t Case I. Since f ( Y N U) m e e t s q t h e r e e x i s t s a sequence of p o i n t s p~E Y n U such t h a t p~--+q as n J" c~. Since Case I is e x c l u d e d , a n d since q is n o n - s i n g u l a r if in a n y f R I (RE W), t h e r e is a t m o s t one R E W such t h a t q m e e t s fiR. W i t h o u t loss of g e n e r a l i t y we can t h e n s u p p o s e t h a t Pn is in s o m e RnE W such t h a t f i R n does n o t m e e t q. T h e r e ac- c o r d i n g l y e x i s t s a p o i n t r~ in flR~ D Y such t h a t r~-+q as n 1' oo. L e t ~ be a t r a n s v e r s a l r a y of Y i n c i d e n t w i t h q. W e can t a k e r~ in ~t. L e t ~n be t h e o p e n arc in )1. s e p a r a t e d f r o m q o n ~t b y r~. T h e n ).~ is in R~ b y L 3.0. Since r~-->q as n r 0% ). is in U a n d hence Y is in U.
C U R V E F A M I L I E S F * L O C A L L Y T H E L E V E L C U R V E S O F A P S E U D O H A R M O N I C F U N C T I O N 11 T h u s (m) is true in Case I I as well as in Case I. S t a t e m e n t (b) follows from (m) a n d (a) of the lemma.
Proo[ o/(c). To show t h a t fl U IE* is simple note t h a t U in (c) is simply connected a n d hence an F-region. Moreover U contains N or S, a n d hence b y Th 3.2 the c o m p o n e n t s of fl U I E* contain neither a meridian nor the carrier of an N- or S-circuit. Th 3.2 t h e n implies t h a t the remaining c o m p o n e n t s of fl U f ~ * do n o t intersect, so t h a t / ~ U I y~* is simple.
If U were n o t concave there would exist an arc h E F * 1U with a limiting end p o i n t z in fl U. L e t z 1 be a p o i n t of h. F o r some n, H n contains Zr H~ excludes z. N o r does flH n contain z, since f l H n is non-singular a n d contains neither N nor S. T h u s f l H n separates z f r o m z 1 a n d so m u s t m e e t h. Since flH n is non-singular this is impossible.
The inner closure ]~ in E * of a set E is defined b y the e q u a t i o n
(3.2) /~ = E N E * - f l E .
E q u i v a l e n t l y t h e inner closure in E* of a set E is the set of all points in E* which possess a n e i g h b o r h o o d in which E is everywhere dense.
L e m m a 3.3. I / R satis/ies Conditions (9 its inner closure in ~*, the complement o/ its closure, and any component o / R also satis/y Conditions (9.
Consider the F - s e t /~. We shall show t h a t /~ satisfies (9 with R. To t h a t end let q be a point of f i r in 22*. T h e F - r a y s in fiR incident with q are in two classes. Class i (i = 1, 2) consists of the F - r a y s incident with i F-sectors in R incident with q. F - r a y s in Class 2 in fl R are n o t in R'. T h e n u m b e r of F - r a y s in fl R incident with q a n d in Class 1 is even (possibly zero) so t h a t the n u m b e r of F - r a y s in fl/~ incident with q is even. If this n u m b e r is zero q is n o t in fl/~. T h u s / ~ satisfies O with R.
The remainder of the l e m m a is readily verified.
w 4. Asymptotes
L e t h be an open arc in F * a n d q a point of h. Sense h. The sensed open subarc of h following q on h will be called an F * - r a y 7r. L e t ~ be given as a 1 - 1 continuous image in 2]* of the interval 0 < t < ~ , with the point Jr(t) corresponding in ~ to t. L e t ~ be an F * - cycle in Y. given b y a m a p p i n g of a circle, on which w = e t~ into 22, so t h a t 9~(0) cor- responds to w = e ~~ a n d 9~(0 + 27r) =~c(O). W e s a y t h a t 7r a n d h are a s y m p t o t i c to q~ in the positive sense of ~ if for some admissible representation of ~r
dist [r (0), ~ (0)] --* 0 [as 0 f oo].
I n discussing a s y m p t o t i c r a y s t h e following l e m m a is f u n d a m e n t a l .
12 J A M E S J E N K I N S A N D M A R S T O N M O R S E
L e m m a 4.1. Let 2 be a transversal. I / a n F * - r a y 7e meets t i n / ) o i n t s Pl, /)2, /)a which are successive on x in the set o/intersections o / ~ and t , then these points are also successive on i in one o / t ' s two senses, and 7~ always crosses t in the same sense.
F o r i ~ ] l e t ~ j be t h e arc of 7~ b o u n d e d b y / ) i a n d / ) i , a n d let 1~ j b e t h e arc of 1 b o u n d e d b y / ) i a n d / ) j . W e show first t h a t / ) 1 , P2, Pa occur in t h e o r d e r w r i t t e n on 1, r e g a r d i n g / ) a , /)2, Pl as t h e s a m e o r d e r on t .
Case I. The ray 7e crosses t in opposite senses at/)1 and P2.
T h e a r c s ~'g12 a n d i21 f o r m a t o p circle. L e t R12 be t h e r e g i o n b o u n d e d b y
21~12121
in ~:a n d n o t c o n t a i n i n g I - i12. B y v i r t u e of T h 2.3, R n i n c l u d a s a pole, s a y S. T h e a r c s 7~2a a n d 132 also f o r m a t o p circle. L e t R2a b c a r e g i o n b o u n d e d b y t h i s c u r v e c h o s e n so as n o t t o c o n t a i n R n . T h e n R23 c o n t a i n s N. A n y o r d e r of/)1, P2, Pa on 1 o t h e r t h a n t h a t w r i t t e n will be s h o w n t o be i m p o s s i b l e .
S u p p o s e t h e o r d e r on i is/)1,/)a, P2 or/)3,/)1,/)2. T h e s e a r e t h e o n l y o r d e r s w h i c h m u s t be cxclu(lcd.. I n t h e s e cases z l a a n d ~al f o r m a t o p circle b o u n d i n g t h e region
R,3 - Z - (R12 U ]~23).
H c r c R13 is s i m p l y c o n n e c t e d , c o n t a i n s no pole, a n d in flR13 , ~,3 m e e t s t13 twice. On M t h e r e e x i s t s a h o m c o m o r p h i c c o v e r i n g of Rla on which ~ a M a n d ~t~.~M m e e t twice. T h i s is c o n t r a r y to Th 2.3. T h u s Pl, /)2, /)3 is t h e o n l y o r d e r p o s s i b l e in I .
Case 11. The ray 7~ crosses I in the same sense at Pl and P2.
S u p p o s e Pl, P~, P2 is t h e o r d e r in 1. T h e n ).t~ a n d ~12 t o g e t h e r b o u n d a r e g i o n R i n t o whose i n t e r i o r 7~ e n t e r s a t / ) 2 . C o n t i n u e d in t h i s sense ~z m u s t l e a v e R b y crossing )~n a t P3 in a sense o p p o s i t e t o t h e crossing a t / ) 1 an(l/)2. On r e v e r s i n g t h e sense of ~ a n d d e n o t i n g /)1, P.z, Pa b y Pa, /)2, Pl r e s p e c t i v e l y , t h e s i t u a t i o n comes u n d e r Case I a n d is i m p o s s i b l e . T h e or(ler P3, Pl,/)2 , n a y be e x c l u d e d in Case I I as c o n t r a r y to t h e J o r d a n S e p a r a t i o n T h e o r e m . T h u s t h e o r d e r on 1 m u s t he/)1, /)2, /)a in a n y case.
I t r e m a i n s to show t h a t all crossings of 1 b y ~ a r e in t h e s a m e sense. W c first show t h a t Case I is i m p o s s i b l e . I n Case I, x r e v e r s e d in sense e n t e r s Rla a t / ) 1 , a n d c o n t i n u e d as a n h E F * m u s t m e e t )[~3 in a p o i n t / ) 0 ; for t h e r e is no p o l e in t h e s i m p l y c o n n e c t e d r e g i o n R13 to which h can t e n d , cf. M J 2 Th 7.2. T h u s P0, P,, P~. a p p e a r in t h i s or(tcr on z b u t in t h e o r d e r P,,/)0, P2 or Pl, P2, /)0 on I~3. H e n c e Case I is i m p o s s i b l e .
I n Case I I t h e crossing of 1 a t / ) 3 is in t h e s a m e sense as a t / ) 1 a n d / ) 2 ; for o t h e r w i s e a r e v e r s a l of t h e sense of ~ w o u l d y i e l d Case I a g a i n .
T h i s e s t a b l i s h e s t h e l e m m a .
W e s t a t e a n e x t e n s i o n whose p r o o f is e s s e n t i a l l y i d e n t i c a l w i t h t h e p r e c e d i n g .
C U R V E F A M I L I E S F * L O C A L L Y T H E L E V E L C U R V E S O F A P S E U D O H A R M O N I C F U N C T I O N 13 L e m m a 4.2. Let 2 be a transversal cut with vertex q. Let h be an element i n F * such that h (~ ~ excludes q. I/7~ is an F * - r a y i n h'the conclusion o / L e m m a 4.1 holds as stated.
T h e exclusion of q f r o m h N 2 i n s u r e s t h a t each p o i n t of h N 2 is a n a c t u a l crossing of 2 b y g. All other p o i n t s of 2 are n o n - s i n g u l a r .
E a c h of t h e regions of E i n t o which t h e carrier of a n i n n e r cycle divides E will be called a side of ~ or of [~]. T h e two sides of ~ m a y be d i s t i n g u i s h e d as t h e north or south side of according as the side c o n t a i n s N or S. G i v e n a n N - or S-cycle t h e region H b o u n d e d b y I~[ which c o n t a i n s S or N r e s p e c t i v e l y will be called t h e s o u t h or n o r t h side of ~.
The construction o / t h e region J . L e t 7~ be a n F * - r a y w i t h a positive l i m i t p o i n t pC•*.
W e shall c o n s t r u c t a n F - r e g i o n J such t h a t 7~ is a s y m p t o t i c to a c u r v e carried b y f l J . L e t 2 be a t r a n s v e r s a l r a y i n c i d e n t w i t h p a n d such t h a t ~ i n t e r s e c t s ~ i n a n i n f i n i t e sequence of p o i n t s
(4.1) PI, P2, Pa . . . .
w i t h l i m i t p o i n t p. Suppose t h a t t h e i n t e r s e c t i o n s of r~ w i t h 2 a p p e a r on r~ i n t h e order (4.1). I n accordance w i t h L 4.1 t h e p o i n t s (4.1) a p p e a r on )~ i n t h e same order.
F o r k = 1, 2, ..., let ~k be t h e arc PkPk+l of ~ a n d 2k the arc pkpk+l of 2. L e t Jk be t h e open region b o u n d e d b y the t o p circle zk2k chosen so as n o t to c o n t a i n p. T h e n
J l c J2 c Ja c . . . . B y v i r t u e of T h 2.3
gl
m u s t c o n t a i n a pole, s a y S. L e tJ = U n i o n Jk (k = 1, 2 . . . . ).
T h e n J is a s i m p l y c o n n e c t e d region which c o n t a i n s S. N o w Z - J k m e e t s N a n d hence N E E - J = N ( E - J k ) ( k = 1 , 2 . . . . ).
k W e c o n t i n u e with a proof of t h e following.
(i) There is no singular point i n ~ /ollowing p~ i n ~.
Suppose z were such a s i n g u l a r p o i n t . T h e r e t h e n exists a n 0r F w i t h z as a n i n i t i a l p o i n t a n d :r N ~ = 0. L e t h be a n F * - r a y c o n t i n u i n g ~. Since z is n o t i n t h e t r a n s v e r s a l )1.
a n d follows p~ in z , a s u f f i c i e n t l y r e s t r i c t e d o p e n arc of h w i t h e n d p o i n t z i n ~ is i n J~+l - J ~ = Q~
for some n > o. Observe t h a t
flQn = U n i o n zt~ 2nZt~+ 12~+I, a n d t h a t t h e set Q* = Q~ - p~+l is s i m p l y c o n n e c t e d .
Since Q~ is s i m p l y c o n n e c t e d a n d c o n t a i n s n e i t h e r N n o r S, h m u s t m e e t flQn i n a p o i n t z' following z o n h. T h e r e are several a priori possibilities for t h e l o c a t i o n of z a n d z' i n flQn b u t i n each ease one infers t h e existence of a n i n n e r cycle of F * i n Q~*, or else of a s u b a r e
14 J A M E S J E N K I N S A N D M A R S T O N M O R S E
b c Q* of a n e l e m e n t in F * such t h a t b i n t e r s e c t s 2 in m o r e t h a n one p o i n t . Since t h i s is i m p o s s i b l e (i) follows.
(ii) f J is the set ~2 o/positive limit points o/7~, so that the m a x i m u m distance o / p o i n t s o/7~n/rom f l J tends to zero as n ~ ~ .
S t a t e m e n t (ii) is a c o n s e q u e n c e of a n e l e m e n t a r y l e m m a , n a m e l y t h a t f J is t h e set of all p o i n t s which are l i m i t p o i n t s of s e q u e n c e s z,~, n = ], 2 . . . . in w h i c h z~ is in f J ~ . T h i s l e m m a p r c s u p p 0 s e s t h e i n c l u s i o n r e l a t i o n s J z c J2 c J a ... a n d t h e d i s j o i n t n e s s of t h e t o p circles f J ~ , n = 1, 3, 5 . . . One n o t e s t h a t f J ~ s e p a r a t e s f J n - 2 f r o m f J n + 2 . S t a t e m e n t (ii) follows on r e c a l l i n g t h a t f J n is t h e u n i o n of A n a n d g~, a n d t h a t t h e d i a m e t e r of ;t~
t e n d s to zero as n ~ ~ .
(iii) J is a concave F-region whose curve boundary is an F*-cycle.
I f t h e b o u n d a r i e s of t h e r e g i o n s J ~ were t o p circles in F , J w o u l d s a t i s f y C o n d i t i o n s O on a n F - r e g i o n , b y L 3.2(c). W e can, h o w e v e r , a l t e r F in a n F - s e c t o r Y c o n t a i n i n g t h e t r a n s v e r s a l r a y ;t u s e d in c o n s t r u c t i n g J so t h a t fl J,~, n = 1, 3, 5 . . . . b e c o m e s a t o p circle in a new f a m i l y F ' r e p l a c i n g F . T h i s a l t e r a t i o n shall l e a v e F ] ( E - Y) u n a f f e c t e d , a n d can be m a d e in Y in a n o b v i o u s m a n n e r l e a v i n g Y a n F ' - s e c t o r . Cf. p r o o f of L 7.1 in M J 2.
W e conclude t h a t J satisfies C o n d i t i o n s O, is concave, a n d t h a t f J l Y , * is s i m p l e [L 3.2].
Since J is s i m p l y counecte(1 a n d f J c o n t a i n s m o r e t h a n one p o i n t , i t follows t h a t J is a n F - r e g i o n . Since J c o n t a i n s a pole, f J carries no N - or S - c i r c u i t s n o r N S - or S N - c u r v e s . I t s c u r v e b o u n d a r y as g i v e n b y T h 3.2 r e d u c e s to a n F * - c y c l e .
T h i s e s t a b l i s h e s (iii).
T h e p r i n c i p a l t h e o r e m of t h i s s e c t i o n follows.
T h e o r e m 4.1. I / a n F * - r a y 7~ has a positive limit point p E Z * then 7~ is asymptotic to an F*-cycle q;. Moreover, 7e does not intersect qz, and q) is concave toward the side o/q~ which con- tains ~.
T h e r a y ~ d e t e r m i n e s a n F - r e g i o n J as j u s t c o n s t r u c t e d . B y (iii) f J carries a n F * - cycle ~ a n d we shall show t h a t T =~0 • satisfies Th 4.1. I n a c c o r d a n c e w i t h T h 3.1 ~ m a p s t h e circle C ([w I = 1) i n t o Big in a m a n n c r w h i c h is 1 - 1 o v e r t h o s e o p e n a r c s of C whose i m a g e s a r c in f l J ] Z * . W e s u p p o s e t h a t t h e p o i n t w = e *e in C c o r r e s p o n d s t o t h e p o i n t
~(0) in f J . I d e n t i f y t h e p o i n t p of T h 4.1 w i t h t h e p o i n t p u s e d in t h e c o n s t r u c t i o n of J a n d s u p p o s e t h a t p = ~ ( 2 n ~ ) , n = 0, :t: 1, ~ 2 . . . R e c a l l t h e t r a n s v e r s a l r a y ;t i n c i d e n t w i t h p, a n d t h e p o i n t s Pl, P~, ... on ;t N g u s e d in t h e c o n s t r u c t i o n of J .
W e s t a t e t h e following.
(~) The Frdchet distance on Z of the subarc PnPn+i of ~ /rom the arc ~ [ ( 0 _<0 _< 2~e) tends to zero as n ~ oo [for p r o p e r choice of ~0 as ~ i ] .
C U R V E F A M I L I E S F* L O C A L L Y T H E L E V E L C U R V E S O F A P S E U D O H A R M O I ~ I C ~FUNCTIOlff 1 5
T h e proof of (9) starts with (ii) a n d is almost identical with t h e proof of Th 4.1 of M J 3.
I t will be omitted. G r a n t i n g (9) one can t h e n m a p the arc PnPn+l of ~ in a h o m e o m o r p h i c m a n n e r o n t o the interval 2n:~<_O _ < 2 ( n + 1)z, n - 1 , 2 . . . . with zr(0) a single-valued continuous image of 0 for all such 0, a n d such t h a t
dist [~(0), ~ ( 0 ) ] - + 0 [as 0 q' ~ ] .
This representation of ~ f r o m Pl on is 1 - 1 a n d so admissible, a n d 7e is accordingly a s y m p - totic to %
B y L 3.2(c), J is concave so t h a t ~ is concave t o w a r d s t h e side of ~0 which contains 7e.
This completes the proof of t h e theorem.
Corollary 4.1. N o point o[ an open arc hE F * is a positive or negative limit point o t h.
This is i m m e d i a t e if h is n o t an a s y m p t o t e , a n d true for a s y m p t o t e s since a s y m p t o t e s c a n n o t intersect the concave F * - c y c l e s which are their positive or negative limit sets.
This corollary also follows from L 4.1 b y a suitable a r g u m e n t .
w 5. Concave annuli .,4 (T1, T~)
We shall consider open ammli A (V1, ~2) in E* each b o u n d e d b y two non-intersecting F*-cycles, ~1 a n d q~2, and, in the case in which the a n n u l u s is concave, give a complete description of elements of F * in the annulus. We proceed with three lemmas.
L e m m a 5.1. A sensed hE F * which is asymptotic i n its positive sense to an F*-cycle q~
can be asymptotic in its negative sense neither to q) + nor to q) - .
L e t h be divided b y a point p into two r a y s ~ ' a n d ~". L e t )~ be a transversal t e n d i n g to a point q of ~ f r o m the side of ~0 which contains h. If t h e l e m m a were false n ' a n d 7e"
would intersect 2 in sequences of points p'n a n d p~', n = 1, 2 . . . . respectively, tending to q in ~t as n ~ ~ , with the order
(5.1) .... p~, p;, q, p',', p~',
. . .on h. These points c a n n o t a p p e a r in the order (5.1) on ~t as t h e y should b y L 4.1, a n d we infer the t r u t h of L 5.1.
Corollary 5.1. I n the set o/ F * - r a y s issuing /rom a given point p E E * and asymptotic either to q~ § or q~ - , where q~ is an F*-cycle, there is at most one F * - r a y .
L e m m a 5.2. I t ~1 is an inner cycle and q~2 a second inner cycle or a concave N - or S-cycle, then q~l is concave toward q~2.
16 J A M E S J E I ~ I K I N S A N D M A R S T O N M O R S E
The cycle ~1 does n o t intersect ~2. This follows f r o m Cor 2.1, if ~ is an inner cycle, a n d f r o m t h e c o n c a v i t y of ~2, if ~2 is an N- or S-cycle. F o r definiteness suppose t h a t
% is on the n o r t h side of ~l-
If the l c m m a were false there would be an F * - r a y 7~ with initial p o i n t r in ~ , entering A (~1, F2) at the p o i n t r a n d n o t meeting ~ again. This r a y c a n n o t intersect W~ if ~2 is an inner cycle [Cor 2.1], or if ~2 is S-concave. I t follows f r o m T h 4.1 t h a t ~ is a s y m p t o t i c to an F * - e y c l e Fa (possibly ~2) in C1 A (~1, ~ ) . L e t 2 be a transversal in the south side of ~3 tending to a point p ~ % . T h e n 7~ will intersect )~ in an infinite sequence of points pl, p~, ...
tending to p as a limit point a n d appearing on ~ in the order w r i t t e n [L 4.1]. L e t
(5.2) ~(r, P2), Jt (Pc, Pl), ~(Pl, r)
be respectively subarcs of ~ f r o m r to P2, of ,~ f r o m P2 to Pl, a n d of ~ f r o m Pl tb r, f o r m i n g a sequence b of arcs joining r to itself. L e t bM be an arc covering b on M. T h e end points of bM cover r b u t are n o t coincident. T h e y can, however, be joined on M b y an arc covering I qJll a finite n u m b e r m of times to f o r m a t o p circle gM Oil M. (That m - 1 is true, b u t n o t necessary for the proof.)
L e t ~tM be the covering of )~ which meets gM. T h e n g M - 2M a d m i t s an extension on M as an clement hM in F ~ . This is impossible since h~ meets ~M in t w o points. [Th 2.3.]
We infer the t r u t h of the lcmma.
L e m m a 5.3. I n a concave a n n u l u s B between two F*-cycles there can be no singular point P.
First note t h a t a n y inner cycle 9~ in B m u s t bc non-singular. F o r it follows from L 5.2 t h a t q~ m u s t be b o t h N- a n d S - c o n c a v e a n d hence non-singular.
Suppose t h e n t h a t L 5.3 is false in t h a t P exists. There t h e n exist at least four F - r a y s issuing from P. T h e c o n t i n u a t i o n s as elements in F * of none of these F - r a y s can c a r r y an inner cycle, since such an inner cycle wouhl be singular. There are t h u s at least four F * - r a y s issuing f r o m P. N o n e of these F * - r a y s can meet fl B or h a v e a limiting end point at N or S in fiB, since B is concave. I t follows f r o m Th 4.1 tilat each such F * - r a y m u s t be a s y m p t o t i c to an F * - c y c l e in /~. Moreover two different F * - r a y s issuing f r o m P are a s y m p t o t i c to F * - c y c l e s 95 a n d ~o 2 with different carriers [Cor 5.1].
T h e cycles ~01 a n d ~02 are concave t o w a r d their respective sides containing P, so t h a t P is in A (T1, ~2); for if cpi s e p a r a t e d ~0s f r o m P [i, ~ = (1, 2) or (2, 1)] t h e n t h e a s y m p t o t i c r a y from P to ~0j would intersect ~i, c o n t r a r y to t h e c o n c a v i t y of ~0 i t o w a r d P . A n y t w o remaining r a y s issuing f r o m P determine an a n n u l u s A (~a, ~ 4 ) c A (q01, ~2). F r o m t h e reciprocity of t h e pairs (~01, ~02) a n d (Ta, ~4), we infer t h a t A (T1, q02)= A (~0a, q%). H e n c e
CURVE FAMILIES F * LOCALLY THE LEVEL CURVES OF A PSEUDOHARMONIC FUNCTION 17 with proper notation iT1] = I%I and ]iP~] = 19%1. Two of the four rays m u s t then be a s y m p t o t i c to two cycles with the same carrier, contrary to Cot 5.1.
Hence P does not exist and the l e m m a is true.
Types o[ asymptotes in A (fol, io~). Suppose t h a t A (/% iv2) is concave and includes no inner cycle. An h E F * in A(~pl, Iv2) is an a s y m p t o t e in both its senses since A(fpl, q)2) is concave. I t follows from Cor 5.1 t h a t in one of its senses h is a s y m p t o t i c to iv1 • and in the other to iv2 ~=. If h is a s y m p t o t i c to tP~ in one sense and to Iv2 in its other sense then h is said to be of a s y m p t o t i c t y p e [iVl, iP2] = [f02, iv1] 9 The four possible a s y m p t o t i c types of h are
[i~ +, ~2 +
], [ ~ + , iv2- ] ,
[rh - , ~ + ], [iP~ - , !v~ - ] .Two elements h and h' of F * in A (f01, io2) m u s t be of the same a s y m p t o t i c type. Other- wise h and h' would intersect in a point P. The existence of P becomes clear on considering the elements hM and hM covering h a n d h' respectively on M. The intersection P cannot exist b y L 5.3.
We shall complete the analysis of annuli A (~vl, iv2) in w 8.
w 6. N-caps, S-caps
I n the decomposition of E into basic regions of a nature dictated by F * one comes naturally to N-caps and S-caps. For the purpose of defining these caps and for m a n y other purposes we shall abbreviate the phrase "inner closure ill E* of the union" by the word symbol ]~nion. With this understood an N - c a p [S-cap] is the l~nion of all non-singular N-loops [S-loops]. This definition requires justification and elaboration.
There is at most a countably infinite number of elements in F with singular end points.
Through each neighborhood in E* there accordingly passes a non-singular h E F * . There are also examples of families F * such t h a t through each neighl)orhood in Z* there passes a singular hE F*. Thus non-singular elements in F* are in fact everywhere dense, while singular elements in F * may be everywhere dense. If in particular a neighborhood is in the interior of an N-loop [S.loop], then each non-singular hE F * meeting tbis neighborhood carries an N-loop [S-loop]. This fact will simplify subsequent proofs.
Beside the question as to the vanishing of N-caps [S-caps] there is the question as to whether N-caps [S-caps] are bounded from S [N]. This leads to the natural separation of the cases in which N-loops [S-loops] are or are not bounded from S [N].
A first theorem follows.
Theorem 6.1. I[ N-loop8 are not bounded ]rom S there exists an F-region Q which is an F*-set, whose curve boundary meets Y~* in an N S . and an SN-curve and in disjoint N- and S-loops at most countable in number.
2--533807. Acta Mathematica. 91. I m p r i m 6 le 18 m a i 1954.
18 J A M E S J E N K I N S A N D M A R S T O N M O R S E
There exist non-singular N-loops bearing points arbitrarily near S. I t follows from L 3.1 that there is an infinite sequence ~1, ~2 . . . . of nonsingular N-loops such t h a t
(6.1) I ( ~ 1 ) c I ( ~ 2 ) c ....
while dist [8, ~n]-+0 as n ~ ~o. We shall show t h a t the set Q = Union I(~n), n = 1, 2 . . . . satisfies the theorem.
The set Q is simply connected and flQ contains N and 8. Q satisfies Conditions O by L 3.2(b), and so is an F-region. I t is an F*-set since a union of F*-sets. I t follows from Th 3.2 t h a t the closed curve b o u n d a r y of Q meets Z* in an N8- and SN-curve, and in disjoint N- and S-loops at most countable in number.
Corollary 6.1. I / N - l o o p s are not bounded/rom 8 there exists at least one meridian in F*.
Th 6.2 is similar to Th 6.1, with g in Th 6.2 replacing S in Th 6.1. The prdof is similar.
Theorem 6.2. Let g be a top circle in F*. I / N-loops are not bounded/rom g there exists an t~-region which is an F*-set, which is in the north side o/ g and whose boundary meets N and g.
.One defines Q as in the proof of Th 6.1, except that here dist [g, ~ , ] - + 0 as n I' ~ . One continues as in the proof of Th 6.1.
N-caps, S-caps. We have already defined an N-cap [S-cap]. One can equivalently define an N-cap UN as the (~nion of the interiors of all non-singular N-loops. If there are no N-loops we understand that UN = 0. S-caps Us are similarly defined. We tcrm UN [Us]
bounded if bounded from 8 [N], otherwise unbounded. I t is possible that UN or Us m a y equal Z*.
Maximal N-cycles, S-cycles. B y the interior I ~ of an N-cycle ~ is meant the union of the interiors of all the N-loops carried by I~1- An N-cycle ~ will be termed maximal if I r ~ I ~ whenever ~ is an N-cycle. There m a y be no maximal N-cycle ~, but when one exists the N-cap UN * 0 and fi UN = I~l" Maximal S-cycles are similarly defined.
Corollary 6.2. I / N - l o o p s are not bounded a w a y / r o m an inner cycle % then ~ is the inner cycle o/ an N-circuit ~p.
Let I N ~ be the north side of ~. According to Th 6.2 F * [ IN~0 includes an open arc h with end points on ]~0[ and N respectively. Then [~01 U h carries an N-circuit ~p with the sense of 90, so t h a t ~ is the inner cycle of y).
Cor 6.2 suggests a major theorem.
Theorem 6.3. (i). I] UN is a bounded, non-empty N-cap, fl(UN U N ) is the m i n i m u m carrier either o / a maximal N-cycle q), or o / a n inner cycle yJ not N-concave.
C U R V E F A M I L I E S F * L O C A L L Y T H E L E V E L C U R V E S O F A P S E U D O H A R M O N I C FU:NCTIOI~ 19 (ii). Conversely, a m a x i m a l N-cycle q) or an inner cycle ~ not N-concave bounds U~ U N , where U N is a non-empty N - c a p b o u n d e d / t o m S.
(iii). A m a x i m a l N-cycle q) and an inner cycle y~ which is not N-concave cannot coexist.
When q~ exists q: - is the only other m a x i m a l N-cycle, and when yJ exists ~v - is the only other inner cycle which is not N-concave.
Proo/ o/ (i). W e begin b y establishing (m) a n d (n).
(m). I] fl U N carries an inner cycle ~v, fl(UN [J N ) = Iy~], and y) is the inner cycle o / a n N-circuit.
The south side I s y~ of ~ does n o t m e e t UN, since no non-singular N - l o o p meets yJ.
Thus UN c IN y~. B y Cor 6.2 ~ is t h e inner cycle of an N - c i r c u i t 9- As such ~ c a n n o t be N - concave. E a c h non-singular element in I N ~ is an N - l o o p since there is an open arc in
F*[Xg~v
joining F to N. Hence UN U N = IN~P a n d f l ( U g U N ) = ]~f].(n). I / U N ~ 0 and i / f l U N carries no inner cycle, fl U N is the m i n i m u m carrier o / a m a x i m a l N-cycle.
Set H = X - C1 UN a n d let L be the c o m p o n e n t of H which includes S. W e shall show t h a t L is an F-region. Since UN is an F-set, L is all F-set. Since /3~ is connected, L is simply connected. Finally L satisfies Conditions O on F-regions with H a n d UN [L 3.3]. N o w f l L carries no inner cycle a n d hence no N-circuit. B y Th 3.2, flL m u s t l)e the m i n i m u m carrier of an N - c y c l e q~. This N-cycle m u s t be a m a x i m a l N-cycle; otherwise L would' m e e t the interior of some N-loop, col~trary to its definition. The interior I ~ of a m a x i m a l N - c y c l e
is an N - c a p UN. Hence flUN :: ~jV[ a n d ( n ) i s established.
S t a t e m e n t (i) follows from (m) a n d (n).
P r o o / o / ( i i ) . We first (~stal)lish (a).
(a). A ~ inner cycle y, which is not N-concave is the inner cycle o / a n N-circuit ~.
If y~ is n o t N-concav(,, an F * - r a y 7r exists in the n o r t h side of y~ with an initial point in yJ. This r a y m u s t h a v e a limiting end point at N. Otherwise zr would bc a s y m p t o t i c to an S-concave F * - c y c l c Y~I n o r t h of yJ. [Th 4.1], a n d yJ wouht be N - c o n c a v e b y L 5.2, c o n t r a r y to hypothesis. I t is clear t h a t ty~[ U 7r carries an N - c i r c u i t r / w i t h the sense of F, so t h a t y~
is the inner cycle of ~]. This establishes (a).
If y~ exists a n y non-singular clement in I(~/) is an N - l o o p b y (a), a n d all non-singular N-loops are n o r t h of y~. I t follows t h a t fl(UNU N ) = V . If ~ exists fl(UNU N ) = l~[, as already noted. I n b o t h cases UN =~ 0.
P r o o / o / ( i i i ) . There is precisely one N - c a p UN, in accordance with its definition. N o w fl(VN t ) N ) = ]~0] or [~v I b y (ii), when 99 or ~v exists. These possibilities are m u t u a l l y ex-
clusive since fl UN is unique. S t a t e m e n t (iii) follows.
T h e o r e m 6.3 has an obvious c o u n t e r p a r t for S-caps.
