The case of loop coverage arises, by definition, when points on N- or S-loops are everywhere dense in ~*. Clearly a necessary and sufficient condition for loop coverage is t h a t
~ n i o n [UN, Us] =Y,*.
When there is loop coverage it will appear t h a t there is at least one meridian in F*. Cf.
Cor 6.1. When there is at least one meridian the decomposition of E* can be studied under the case of loop coverage and the case of no loop coverage. We here study loop coverage.
B y a maximal N-loop is meant a n y N-loop ~0 such that Iq~ ~ Iv/ whenever ~p is an N-loop with I v N I~0 * 0. A maximal S-loop is similarly defined.
We need further information regarding unbounded N-caps UN.
Theorem 9.1. I / UN is not bounded/rom S and :*= E*, then each component R of UN is either i, the interior o / a maximal N-loop or, ii, an F-region bounded by N U S, by two disjoint meridians and at most countably many disjoint maximal S-loops, iii, a region bounded by a maximal S-cycle and by N.
(a) Two components o/ U N o/ types i or ii have disjoint boundaries in E*.
(b) There is at least one component o/ type ii or iii. A n y component o/ UN o[ type iii equals U ~. The number o/components o/type ii is/inite.
28 J A M E S J E : N K I N S A:ND M A R S T O N M O R S E
L 3.2 implies t h a t each component R of UN is an F-region. As given b y Th 3.2 fiR does not carry an N or S-circuit since UN is not bounded from S. An R is then of t y p e i if bounded from S, of t y p e iii if N is isolated in fl R, of t y p e ii otherwise. The m a x i m a l i t y of the loops follows from the definition of UN.
Intersection o/ component boundaries. Two components of UN cannot have an open boundary arc in common since UN is an inner closure in Y~*. If R 1 and R 2 are two components of UN of types [i, ii], [i, i] or [ii, ii] then fiR 1 and fiR 2 cannot meet in a point in Y~*; other- wise f i r 1 UflR 2 would carry an N-loop with interior R o such t h a t l~nion [R0, R1, R2]
would be connected and in UN, contrary to the nature of R 1 and R2 as components of UN.
Number o/ components. If there were no component of t y p e ii or iii, UN would be bounded from S, contrary to hypothesis; t h a t a n y component of t y p e iii is UN itself follows from the relation
UN :D R ~ Z * - ~TsxD UN.
The number of components R of UN of t y p e ii is finite; for there exists in each such R an N-loop with diameter exceeding z / 2 and the n u m b e r of such N-loops in different com- ponents R is finite b y L 3.1.
This completes the proof of the theorem.
By an a r g u m e n t similar to t h a t used in the last p a r a g r a p h of the proof one can show t h a t the m a x i m a l n u m b e r of meridians in any collection of disjoint meridians is finite in the ease of loop coverage.
On setting
(9.1) B = fl U N N fl Us
one obtains the following theorem.
Theorem 9.2. I n the case oJ loop coverage, the ]ollowing is true.
(a) I[ UN is bounded ]rom S and Us ]rom N, B = ~ U N - - N = f l U s - S is a top circle in F*.
(b) I] UN is not empty and bounded/rom S, but Us is not bounded/rom N, B = fl UN is the carrier of a maximal N-cycle. A similar statement holds interchanging N and S.
(c) I / UN is not bounded/rom S nor U s / r o m N, the components o/ BlY~* are simple and disjoint, and include a/inite set (at least two) of meridians, and carry at most a countable set o/maximal N- and S-loops.
Statements (a) and (b) follow from Th 6.3 and s t a t e m e n t (c) from Th 9.1.
Primitives. We shall give another decomposition of ~:* in t e r m s of a ~ n i o n of certain elementary regions to be termed primitives. These primitives will enter into decomposi-
C U R V E F A M I L I E S ~ * 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 29 tions b o t h in t h e case of loop coverage, a n d in the case where at least one meridian exists a n d there is no loop coverage.
De[inition. L e t q01, q2 . . . . be a sequence of disjoint non-singular N-loops such t h a t
(9.2) I ~ 1 c I ~ 2 c I ~ a = . . . .
Then U n i o n I ~ n will be called an N-element a n d d e n o t e d b y [~0]. A n N - e l e m e n t which is n o t a proper subset of a n y other N - e l e m e n t is called an N-primitive. S-primitives [q~] are similarly defined.
W e shall establish a n u m b e r of propositions which lead up to a decomposition of
UN [Us]
into a ]~nion of disjoint N - p r i m i t i v e s [S-primitives] c o u n t a b l e in n u m b e r . The essence of the analysis lies in the i n t r o d u c t i o n of a partial order a m o n g N- or S-elements, and, for ordered subsets of N- or S-elements, in the r e d u c t i o n of this order to a numerical basis.W e s a y t h a t two N - e l e m e n t s are ordered if one is included in the other. We similarly order interiors I ~ of N-loops y). Strict inclusion of a set A in B will be d e n o t e d b y the rela- tion A < B or B > A. I f ~ a n d ~p are non-singular N-loops a n d I ~ N Iy) * 0 I ~ a n d Iy) are ordered. We e x t e n d this fact as follows.
(~) I[ N-elements [(p] and [y)] intersect, [q~] and [~] are ordered.
If [q0] N [~o] * 0 t h e n for suitable integers r a n d s, Iq% N Iy)~ -~- 0, a n d hence I ~ n N I~,,~ * 0 for n ~ r, m ~ s. Hence t h e set of all loop interiors of the f o r m Iq~n, n ~ r, Iy)m, m ~ s, is ordered. F r o m this set one can f o r m an N - e l e m e n t [~] such t h a t [~] ~ [~] a n d [~] ~ [y)].
I t is clear t h a t either [~] = [~], or [$] = [~], or t h a t both of these equalities hold. S t a t c m e n t (a) follows.
(~) I[ [q0] and [~p] are ordered N-elements with diameters D[(p] and D[lp] respectively, then D [~0] > D [~] i / a n d only i] [q~] > [~p].
Suppose t h a t I~% a n d Iy),~ are ordered, a n d let d(%~) a n d d(F,,) be the diameters of
~ a n d yzn respectively. I t is clear t h a t d(%~) > d(y),,) if a n d only if Iffn > I~,,,.
{1) If [~0] > [F], the a b o v e N - e l e m e n t [~] is such t h a t for some integer t, a n d for n ;:: t,
~n is in the set [~1, ~2 . . . . ] a n d I ~ t D [y)]. Hence D [ ~ ] = D[~] > d(~t) -~ D[y~].
(2) I f D[T] > D[y)], [~] has this same p r o p e r t y so t h a t [~] > [~].
(y) I/[q~] and [y)] are ordered N-elements, D [(p] = D [~] i[ and only i/[q~] = [y~].
This is an i m m e d i a t e consequence of (~).
W e state two basic l e m m a s i n d e p e n d e n t of the h y p o t h e s i s of loop coverage.
L e m m a 9.1. The Union V o/ N-elements [S-elements] in any ordered class K o/ N- elements [S-elements] is an N-element [S.element].
30 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
I t is sufficient t o c o n s i d e r t h e case of N - e l e m e n t s . S e t zJ = s u p D[tE] I ( [ ~ ] E K ) a n d d i s t i n g u i s h t w o cases as follows.
Case I. F o r s o m e [yJ]EK, D[~v] = A. I n t h i s case i t follows f r o m ([~) a n d (y) t h a t [~p]
i n c l u d e s e v e r y [~v] E K a n d h e n c e [yJ] D V. B u t [~v] is in K so t h a t V ~ [~v]. H e n c e [y~] = V a n d L 9.1 follows.
Case I I . :Not Case I. I n Case I I t h e r e is a sequence [~vr], r = 0, 1 . . . . of N - e l e m e n t s in K , such t h a t D[~v ~] increases s t r i c t l y as r ~ oo a n d t e n d s t o A as r f' oo. T h e n
(9.3) [~0] < [~vl] < [~v~] < . . . [ b y ([~)].
F o r n successively 1, 2, 3 . . . . one can choose a F~ in t h e set [~l n, ~v~ n, ...] such t h a t
a n d h e n c e T h u s (9.4) so t h a t
I~v~ > [~,~-l]
I ~ l < I ~ 2 < IyJ a < . . . .
[y)] > I~p~ > [~v "-~] (n = 1, 2 . . . . )
D[~f] ~ D[q) ~-~] In = 1, 2 . . . . ]
a n d h e n c e D[yJ] ~ A. B u t D[v2] ~ A b y v i r t u e of t h e d e f i n i t i o n of J . W e c o n c l u d e t h a t D [W] = A. B u t c l e a r l y V = [~p], a n d L 9.1 follows.
Corollary 9.1 The union K el all N-elements [S-elements] which meet a given N-element [S-element] is an N-primitive [S-primitive].
T h e e l e m e n t s in K each m e e t a g i v e n e l e m e n t [~], so t h a t K is t h e u n i o n of a n o r d e r e d set of N - e l e m e n t s [ S - e l e m e n t s ] . T h e c o r o l l a r y follows f r o m t h e l e m m a .
L e m m a 9.2. A primitive R is an F-region. Each component o / f i R in F.* is concave toward R.
T h e region R is a n N - e l e m e n t a n d as such s i m p l y c o n n e c t e d . I t is a n F - r e g i o n a n d a n F * - s e t b y L 3 . 2 . B e c a u s e R is a n F * - s e t t h e c o m p o n e n t s of f i r in ~]* a r e c o n c a v e t o w a r d R.
L e m m a 9.3 (a). Each point in an N.loop [S-loop] is in the closure o / a n N-primitive [S-primitive].
(b) No two primitives intersect.
(c) The number o/ disjoint primitives with diameters exceeding a positive constant is /inite.
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 31 Proo/ o/ (a). W e t r e a t t h e case of a p o i n t P in an N - l o o p ~0. L e t ;t be a transversal r a y in 1 9 a n d incident with P . L e t p , , n = 1, 2 . . . . be a sequence of p o i n t s a p p e a r i n g on +~
in t h e order Pl, P2 . . . . a n d tending to P as n I' co. W e can suppose t h a t each p o i n t Pn is chosen so t h a t the N - l o o p q)n meeting p~ is non-singular. T h e n ~ fl ;t =Pn [L 3.0] a n d P is in E ~ . I t follows t h a t
1 9 1 c 1 9 2 c ...
so t h a t [~] is an N - e l e m e n t a n d C1 [q~] contains P. A c c o r d i n g to Cor 9.1 [~0] is in an N- primitive.
The case of a p o i n t P in an S-loop is similar.
Proo/ o/ (b). A n N - p r i m i t i v e [~] c a n n o t m e e t an S - p r i m i t i v e [y;]; otherwise some non- singular N - l o o p ~% would be in t h e interior of some non-singular S-Ioop YJm. This is clearly impossible. N o r can an N - p r i m i t i v e [9~] m e e t a different N - p r i m i t i v e [y~]. F o r [~] a n d [y~]
would t h e n be ordered [cf. (~)], a n d be equal, since b o t h are m a x i m a l N-elements. State- m e n t (b) follows.
Proo/ o/ (c). I n each primitive with d i a m e t e r exceeding c > 0, there is a loop with dia- m e t e r exceeding e and two such loops in (lisjoillt primitives would h a v e disjoint interiors.
T h e n u m b e r of such loops is, however, finite [L 3.1] a n d (c) follows.
L q.3 yields the following theorem.
Theorem 9.3. There is at most a countable number o/ N-primitives [S-primiti~;es] which meet aa N-cap UN, [S-cap Us], and UN [Us] is the ~niou o/ these primitives.
Corollary 9.2. In the case o/ loop coveraye there is at most a cr>untable number o/ primitives in G*, and Z* is the ~nion o[ these primitives.
w 10. F-guides
A p s c u d o h a r m o n i c function with the open arcs of F as level lines is strictly increasing or.decreasing along a transversal. The existence of simple arcs on M which arc finite se- quenccs of transversc arcs will t u r n o u t to be of the greatest i m p o r t a n c e in t h e s t u d y of p s c u d o h a r m o n i c functions u on M, a n d in answering the question as to the n a t u r e of u as a I u n c t i o n on Y=*, in p a r t i c u l a r in finding p s e u d o h a r m o n i c functions which are single- v a l u e d on Y.* a n d h a v e the open arcs of Y as level lines. F-guides, which we n o w define, are central in this s t u d y .
De/inition. A non-singular arc on •* is t e r m e d m-transverse if t h e union of m con- sequtive transverse arcs. A t o p circle on Y=* is t e r m e d m - t r a n s v e r s e (m > 1 ) i f t h e u n i o n
32 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
of m c o n s e c u t i v e t r a n s v e r s e arcs, a n d 1 - t r a n s v e r s e if e v e r y open s u b a r e is a t r a n s v e r s a l . A n m - t r a n s v e r s e t o p circle s e p a r a t i n g N f r o m S for which m is a m i n i m u m is called a n
F-guide.
T h e existence of a n F - g u i d e is m o s t difficult to establish i n t h e case i n which there exists a t least one m e r i d i a n L, a n d this is t h e case where t h e F - g u i d e is m o s t useful. I n case L exists a n n - t r a n s v c r s e t o p circle g i n Z * which separates N from S, i n t e r s e c t s L i n a single p o i n t , a n d is such t h a t n is a m i n i m u m s u b j e c t t o these c o n d i t i o n s , is called a n FL-guide. A n F L - g u i d e need n o t be a n F - g u i d e , b u t once t h e e x i s t e n c e of a n F L - g u i d e is established t h e existence of a n F - g u i d e follows readily, e v e n i n t h e cases where there is no m e r i d i a n .
Reversing points. A p o i n t of j u n c t i o n P of two successive t r a n s v e r s e arcs whose u n i o n is a n arc g, is called a reversing p o i n t of g if thc sense of crossing of e l e m e n t s ' o f F revcrses a t P . Rccall t h a t P is n o n - s i n g u l a r . I t is clear t h a t t h e j u n c t i o n p o i n t P of two successive t r a n s v e r s e arcs i n a finite m i n i m a l d e c o m p o s i t i o n of a n arc g i n t o t r a n s v e r s e arcs is a revers- ing p o i n t . Otherwise t h e two arcs wouhl form a single t r a n s v e r s e arc a n d g couhl n o t h a v e bccn m i n i m a l l y (tecomposed.
7'he existence o[ an FL-guide. E x c e p t for one p o i n t in L, a n F L - g u i d e g, if it exists, will be. in the region A = ". -- L. T h e region A is the h o m c o m o r p h of a finite z-plane so t h a t t h e results of M J 2 can 1)e applic(l to the f a m i l y F 0 = F I A . Ill M J 2 " b a n d s " p l a y e d a f u n d a m e n t a l role. A b a n d R(N~,), r(~lative to A, is defined as t h e u n i o n of all e l e m e n t s in F 0 which m e e t a right neighb()rhood Nj, in A. As shown in M J 2 a h a n d R ( N ~ ) in A is a n F0-r(,gi(m, a n d has b o u n d a r y (~omi)~)ncnts in ,4 which are simple. ]f E is a set in A it will I)(' necessary to (listinguish b e t w e e n the b o u n d a r y fl E of E r e l a t i v e to Z, a n d t h e b o u n d a r y /~0E of E r e l a t i v e to A.
We b e g i n with two lcmmas.
L [ ' m m a 10.1. A n y two non-singular points Pl and P2 on the boundary f i R o[ a band R
~n fl - Z -- L can be ~oined by (in m-transverse arc g such that g - Pl - P2 is in R and m :~ 3.
A n y two p o i n t s q~ a n d q~ in different e l e m e n t s of F 0 in R c a n clcarly bc j o i n e d b y a t r a n s v e r s e arc in R. B u t the g i v e n p o i n t s Pl a n d Pz can be j o i n e d to p o i n t s ql a n d qz in R a n d n e i g h b o r i n g Pl a n d p,,, resw.ctivcly, b y t r a n s v e r s e arcs ]Q, k.,, in R e x c e p t for Pl a n d P2. 0 n c c a n suppose ql a n d q., so n e a r Pl a n d P2, respectively, t h a t k 1 does n o t m c c t k 2.
Let k be a t r a n s v e r s e arc j o i n i n g q~ to q., i n R. If/c 1 N k = ql a n d ]c 2 N/c = q2 t h e arc g = k l k k 2 satisfies the l e m m a . Othcrwise let ]c~ a n d ]c~ be m a x i m a l i n i t i a l s u b a r c s of ]c 1 a n d / Q , re-
1 r t
spectively, i n t e r s e c t i n g k o n l y i n t h e i r e n d p o i n t s q~ a n d q2, a n d let k' be t h e s u b a r c ql q2 of k. T h e n t h e arc g = k'l ]c' k~ satisfies t h e l e m m a .
CURVE FAMILIES F * LOCALLY THE L E V E L CURVES OF A PSEUDOHARMONIC FUNCTION 33 L e m m a 10.2. I / a meridian L exists an FL-guide g exists.
L e t p be a n a r b i t r a r y n o n - s i n g u l a r p o i n t in L, a n d l e t ~t a n d # be sensed t r a n s v e r s e a r c s j o i n i n g p t o p o i n t s P a n d Q r e s p e c t i v e l y , o n o p p o s i t e sides of L. W e s u p p o s e 2 a n d / ~ so r e s t r i c t e d t h a t )~ (~/~ = 0, ~t A L = p, # ~ L = p.
I t follows f r o m T h 9.1 of M J 2 t h a t t h e r e e x i s t s a f i n i t e set of d i s j o i n t b a n d s (10.I) R1, R 2 . . . R m [m > 1]
of A whose ~ n i o n is a n F - r e g i o n H which c o n t a i n s P a n d Q. L e t P1 a n d Q1 be r e s p e c t i v e l y t h e first i n t e r s e c t i o n of ~ a n d # w i t h fill. T h e p o i n t P1 is n o t n e c e s s a r i l y in fR1, nor Q1 in fl R m.
I f R i a n d Rj, i * j, a r e t w o b a n d s in (10.1) whose l~nion is c o n n e c t e d , / ~ 0 R i ~/~0Rj i n c l u d e s a t l e a s t one c l e m e n t ~ E F o , so t h a t one c a n c o n n e c t R i w i t h R~ b y a n arc which crosses ~ a t one p o i n t only. T h e p o i n t s P1 a n d Q1 c a n a c c o r d i n g l y be c o n n e c t e d b y a non- s i n g u l a r a r c g, in H e x c e p t for Px a n d Q1, a n d m e e t i n g t h e r e s p e c t i v e b o u n d a r i e s
floRi
in a t m o s t a f i n i t e set of s p o i n t s . I f t h e n one chooses g so t h a t s is m i n i m a l , i t follows t h a t /~R i N g, i = 1 . . . m, is e i t h e r t h e e m p t y set or t w o p o i n t s p[ a n d p [ ' a p p e a r i n g in t h i s o r d e r on g. T h e p o i n t s p[ a n d p [ ' can be j o i n e d b y a r - t r a n s v e r s e arc gi (r <__ 3) w i t h g , - p [ - p [ ' c R,. [L 10.I.]
I f p =~P1 a n d p ~ Q 1 , t h e s u b a r c s PP1 of ~t a n d Qlp of it, u n i t e d w i t h t h e arcs g~ in p r o p e r order, give a n n - t r a n s v e r s e t o p circle, w i t h n -: 3 m t ~ 2, m e e t i n g L o n l y a t p. I f P = P1 t h e s u b a r c of PP1 of )~ is n o t n e e d e d .
T h e case in w h i c h p = Q1 is s i m i l a r .
A n n - t r a n s v e r s e t o p circle m e e t i n g L o n l y a t p a n d for which n is m i n i m a l a c c o r d i n g l y exists, a n d t h e l e m m a follows.
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. N o h y p o t h e s i s as t o t h e e x i s t e n c e of a m e r i d i a n is m a d e .
T h e o r e m 10.1. Corresponding to an arbitrary admissible ]amily F defined on F~*, there always exists an F-guide g.
L e t h be a n o n - s i n g u l a r s u b a r c of a n e l e m e n t of F , w i t h e n d p o i n t s P1 a n d P2 in ]E*.
s e x i s t t o p circles g~ a n d g2 in E*, each s e p a r a t i n g N f r o m S a n d w i t h gl N g2 = 0, gl N h =P1, g2 N h = P2. T h e n gl a n d g2 b o u n d a d o u b l y c o n n e c t e d d o m a i n X ~ E*. X is t o p o l o g i c a l l y e q u i v a l e n t to E * u n d e r a m a p p i n g T of X o n t o :E*. U n d e r T, F I X goes i n t o a f a m i l y F ' a d m i s s i b l y d e f i n e d o v e r E * . I n F ' , T (h - P1 - P2) is a m e r i d i a n L ' . F r o m L 10.2 we infer t h e e x i s t e n c e of a n F ' L ' - g u i d e g'. F o r s o m e f i n i t e m, T - l g ' is m - t r a n s v e r s e r e l a t i v e t o F a n d t h e e x i s t e n c e of a n F - g u i d e follows.
To a p p l y t h i s t h e o r e m 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 a r e n e e d e d .
3 - 533807. Acta Mathematica. 9l. I m p r i m 6 lo 19 m a i 1954.
34
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 EL e t p be a n y n o n - s i n g u l a r p o i n t a n d N v a r i g h t n e i g h b o r h o o d of p w i t h c a n o n i c a l coor- d i n a t e s u a n d v. G i v e n r E N v w i t h u ~: 0 a t r, a s e n s e d t r a n s v e r s e a r c g m e e t i n g r will b e s a i d t o be sensed a w a y / r o m p if ] u I is i n c r e a s i n g on 9 as r is a p p r o a c h e d in g's p o s i t i v e sense.
A s i m i l a r d e f i n i t i o n is u n d e r s t o o d o n M .
A construction/or use in L 10.3. G i v e n hfiF*M l e t A a n d B b e n o n - s i n g u l a r p o i n t s in h.
L e t H be one of t h e t w o r e g i o n s i n t o w h i c h h d i v i d e s M . I n H s u p p o s e t h a t t h e r e a r e n > 0 F ~ - r a y s 7[: 1 . . . ;71~ n w i t h e n d p o i n t s in t h e a r c A B of h. S u p p o s e t h e s e r a y s w r i t t e n in t h e o r d e r in which t h e y a r e m e t b y a n a r c k j o i n i n g A t o B w i t h k - A - B c H , a n d m e e t i n g each r a y 7t i in j u s t one p o i n t . L e t ~t a n d p be n o n - i n t e r s e c t i n g o p e n t r a n s v e r s a l s in H i n c i d e n t w i t h A a n d B r e s p e c t i v e l y . I t follows f r o m T h 2.3 t h a t 2 a n d p m e e t n o n e of t h e r a y s .
L e m m a 10.3. I n the preceding con/iguration A can be joined to a n arbitrary point P r i g , or to P = B , by an m-transverse arc g with g - A - P ~ H - ~ - p and such that g is seused a w a y / r o m B when P is in p. The m i n i m u m value o / m is n + 1 when P is i n p, and n + 2 when P = B .
T h e l e m m a is t r u e w h e n n = 0; in t h i s case a m i n i m u m m = 2, w h e n P = B. [Cf. T h 2.3.]
W h e n n > 0 let P~ be a p o i n t in ~r~ f3 H , i = 1 . . . n, a n d set P0 = A . I t is clear t h a t P t - 1 can be j o i n e d t o P~ b y a 2 - t r a n s v e r s e a r c ki whose m a x i m a l o p e n s u b a r c is in t h e set
H * = H - U n i o n (~, p , zt 1 . . . ~tn).
T h e j u n c t i o n p o i n t of t h e t w o t r a n s v e r s e a r c s c o m p o s i n g k i m u s t be a r e v e r s i n g p o i n t [Th 2.3]. M o r e o v e r P n can be j o i n e d t o P E p b y a 1 - t r a n s v e r s e a r c s e n s e d a w a y f r o m B a t P . L e t g be t h e arc j o i n i n g A t o P o b t a i n e d b y u n i t i n g t h e s e arcs. T h e p o i n t s P1 . . . P n in g a r e n o t r e v e r s i n g p o i n t s . T h u s g b e a r s n r e v e r s i n g p o i n t s . T h e s e r e v e r s i n g p o i n t s d i v i d e g i n t o n + 1 t r a n s v e r s e arcs.
L e t g now b e a n a r b i t r a r y m - t r a n s v e r s e a r c s a t i s f y i n g t h e l e m m a . S u p p o s e P E p . L e t Qi be t h e first p o i n t of i n t e r s e c t i o n of g w i t h zt~ a n d K~ t h e l a s t p o i n t . S e t K o = A . T h e s u b a r c s K,n_lQm, m = 1 . . . n, i n t e r s e c t o n l y w h e n successive a n d t h e n o n l y in a c o m m o n e n d p o i n t . T h e y c a n n o t be t r a n s v e r s e arcs, b y T h 2.3, a n d h e n c e b e a r a t l e a s t one r e v e r s i n g p o i n t . T h u s g b e a r s a t l e a s t n r e v e r s i n g p o i n t s so t h a t m > n + 1.
T h e case in w h i c h P = B is s i m i l a r .
Corollary
10.1. A n F-gulde g which meets the interior o] a non-singular loop q~ has precisely one reversing point i n Iq).L e t a n N - l o o p g [S-loop g] i n t o whose s o u t h side [ n o r t h side] t h e r e e n t e r s j u s t one F * - r a y w i t h i n i t i a l p o i n t in g, be t e r m e d S.semi.conv, ave [ N - s e m i - c o n c a v e ] . Cf. M J 2 T h 8.1.
CURVE FAMILIES F * LOCALLY THE LEVEL CURVES OF A PSEUDOHARMOI~IC FUNCTI01~ 3 5
Lemma 10.4 (a). A meridian h which is concave toward one o/ its sides is met by aN F-guide in ~ust one point.
(b). A n F-guide g meets no S-concave or semi-concave N-loop, or N-concave or semi- concave S-loop each point o/ which is the limit point o/ a sequence o/ points on non-singular meridians.
Proo/o/la). The intersections of g with h are isolated on g and hence finite in number.
I f g meets h in more t h a n one point it meets h in at least two points. One then uses L 10.3 to show t h a t g can be modified so as to cross h just once and be an m-transverse arc with m smaller t h a n previously. Since m is supposed to be a minimum for g this is impossible.
Proo/ o/ (b). We suppose (b) false in t h a t g meets an S-semi-concave N-loop ~ satisfying the conditions of the lemma. At the first point p of intersection of g with ~, g crosses ]r Otherwise g will have a reversing point at p and cross some non-singular meridian passing near p more than once, contrary to (a).
Let A and B then be two points, at which g enters I ~ and leaves I ~ respectively, bounding an open subarc g(A, B) of g in I ~ . Consider the case in which the F * - r a y r given in E~o as incident with ~0 has its initial point r in the open subarc ~(A, B) of ~.
There will then be at least one F * - r a y in I ~ incident with r. Let P be a point in an open transversal in g just following g(A, B). ]By L 10.3, g(A, B) carries at least two reversing points. However, there exists a 2-transverse arc gl(A, P), in E~0 except for A in ~, which, substituted for g (A, P), gives a simple closed curve gl in place of g, with a reversing point at A, and at only one other point of gl(A, P), but with no reversing point at P. Thus gl is an F-guide meeting ~ without crossing ~. This we have seen is impossible.
The case in which no F * - r a y 7t is incident with q~ (A, B) is similar. This is the ease which always occurs if q is S-concave. The case of an S-loop is of like character. We infer then t h a t g cannot cross loops conditioned as in the lemma.
The index v (F) o/ F. The number of reversing points in an F-guide g is called the index v (F) of F. I t is independent of the choice of g as F-guide. The following theorem gives an evaluation of v (F).
Theorem 10.2. I~ there is at least one meridian in F*, each reversing point o / a n F-guide g is in a primitive, while each primitive met by g contains just one reversing point o/g. Thus the index v (F) is the number of primitives met by g.
Let P be a reversing point of g. A non-singular element h E F * which meets g in a point q =~ P sufficiently near P meets the two transversal subarcs of g incident with P . W h e n a meridian exists, h as non-singular, must either be a meridian or a loop. B u t tt
36 JAMES J E N K I N S AND MARSTOI~I" MORSE
c a n n o t be a m e r i d i a n b y L 10.4. H e n c e h carries a l o o p ~. T h e n P is in 1 % O t h e r w i s e g w o u l d e n t e r I ~ a t t w o p o i n t s . T h i s is i m p o s s i b l e , for b y L 10.3 a n F - g u i d e c a n m e e t a n o n - s i n g u l a r l o o p in a t m o s t t w o p o i n t s . S i m i l a r l y P is in t h e i n t e r i o r of e a c h of a s e q u e n c e
~1, ~02 . . . . of d i s j o i n t loops whose c a r r i e r s m e e t g in a s e q u e n c e of p a i r s of p o i n t s t e n d i n g t o P as a l i m i t . I f ~vn is p r o p e r l y chosen
I ~ 1 c I ~ 2 c ,..
so t h a t U n i o n I q ) n is a n e l e m e n t c o n t a i n i n g P in its i n t e r i o r . B y Cor 9.1 P is in a p r i m i t i v e c o n t a i n i n g t h i s e l e m e n t . T h a t e a c h p r i m i t i v e m e t b y g c o n t a i n s i u s t one r e v e r s i n g p o i n t follows w i t h t h e a i d of Cor 10.1.
T h e t h e o r e m follows.
w 11. No loop coverage, meridians present
To d e c o m p o s e Y~* p r o p e r l y in t h i s case a new t y p e of c o v e r i n g r e g i o n is n e e d e d t o s u p p l e m e n t N - a n d S - c a p s .
Meridional regions. A m a x i m a l c o n n e c t e d o p e n set R c 2,* in which t h e set of p o i n t s on n o n - s i n g u l a r m e r i d i a n s is e v e r y w h e r e d e n s e is called m e r i d i o n a l . E q u i v a l e n t l y a m e r i d - i o n a l r e g i o n is a m a x i m a l c o n n e c t e d o p e n s e t R c ~ * w h i c h is t h e l ~ n i o n of n o n - s i n g u l a r m e r i d i a n s in R.
I n t h e case a t h a n d tile o p e n set
(11.0) X = Z * - C I ( U N U
Us)
is n o t e m p t y . W e b e g i n with a l e m m a .
L e m m a 11.1. A n y element hE F* which meets X is a meridian.
S u c h a n h c a n n o t be a l o o p since h is n o t in t h e closure of UN or Us, n o r a t o p circle, since i t w o u l d t h e n be c a r r i e d b y a n N - or S - c i r c u i t a n d so b o u n d UN or Us [Th 6.3 ii].
I t c a n n o t c a r r y a n a s y m p t o t i c r a y n since s u c h a r a y , f r o m a c e r t a i n p o i n t on is n o n - s i n g u l a r , c o n t r a r y t o t h e f a c t t h a t n w o u l d m e e t a n y m e r i d i a n in p o i n t s following a n y p r e s c r i b e d p o i n t on ~, H e n c e h is a m e r i d i a n .
A " c o v e r i n g " in M of a m e r i d i a n or N - or S - l o o p in Y,* is called a m e r i d i a n or N - o r S - l o o p in M . O b s e r v i n g t h a t no m e r i d i o n a l r e g i o n can i n t e r s e c t a n N - o r S - c a p , t h e n a t u r a l d e c o m p o s i t i o n of Z * is here a s follows.
T h e o r e m l l . 1 . I n case loop coverage /ails and a meridian exists, Y~* is decomposed as /oUows. S e t B =fl(UN tJ N) N fl(Us D S).
(a). I / B = 0, the set X in (11.0) is a doubly connected meridional region R bounded on the north by fl UN, or by N i/ UN = 0, and on the south by fl Us, or by S i/ Us = O.
CURVE FAMILIE~ F * LOCALLY THE LEVEL CURVES OF A PSEUDOHARMONIC FUNCTION 3 7
(b). I f B ~: 0 each component R of X is an F-region R such that
RM
in M is bounded by two disjoint meridians, whose projections in Z* intersect at most in a point, and by a set (possibly empty) o/disjoint N- or S-loops.(c). The number of components of X is linite.
I t ~ollows from L 11.1 t h a t a n y non-singular element in F * which meets X is a meridian. Hence the components of X are meridional.
Proof of (a). I t is clear t h a t X is bounded as stated, and hence doubly connected.
Proof of (b). Here X is an F-set and satisfies Conditions O with UN and Us. Hence R does likewise [L 3.3]. Since there exists a non-singular meridian there is no N- or S- circuit. Hence the components of fl R in Z * m u s t consist of two meridians h and k, and a set (possibly e m p t y ) of disjoint N- and S-loops. R is in fact an F-region. The meridians h and k intersect at most in a point, since X and hence R is an inner closure. S t a t e m e n t (b) follows.
Proof o/(c). I f there were infinitely m a n y components of X there would be infinitely m a n y meridians in f i X of which no two would intersect in more t h a n a point. There would then be infinitely m a n y components of Un U Us whose closures would meet the equator of Z, and b y virtue of Th 9.3 infinitely m a n y primitives with diameters at least ~. This is impossible b y L 9.3 (c).
This completes the proof of Th 11.1.
L e m m a l l . 2 (i). I / R is a meridional region each element kE F* [ R is carried by a meridian in F*.
(ii). There is at most one element hE F*[ R incident with a given N- or S-loop in fiR, and no such h is incident with a meridian in fl R.
(iii). There is no singular point in R.
Proof o/(i). This follows from L 11.1.
Proof of (ii). Suppose two elements h and k in F * ] R were incident with points p and q in an N- or S-loop ~0 in fl R. Let ~o (p, q) be the arc of [~01 between p and q in case p =~ q, and let ~ (p, q) = p in case p = q. L e t h' and k' be meridians carrying h and k respectively.
I t is then clear t h a t h' U k' U q0 (p, q) carries an N-loop and an S-loop intersecting in q0 (p, q).
Since at least one of these loops meets R this is impossible. T h a t no element hE F * [ R is incident with a meridian in fl R is similarly proved.
Proof of (iii). The denial of (iii) implies the existence of a loop meeting R. Thus (iii) m u s t be true.
38 J A M E S J E N K I N S AI~D M A R S T O N M O R S E
Theorem 11.2 (a). It R is a simply connected meridional region each F.guide g crosses R without reversing point in R and without meeting the loop boundaries o/ R. The union o/
the elements in F meeting g N R is R.
(b). A doubly connected meridional region R exists i/ and only i/ there is an F-guide without reversing point, and in case R exists the union o/all elements in F meeting an F-guide g i s R.
Proo/ o/ (a). B y virtue of L 1i.2 (ii) each N-loop [S-loop] in f i r is S-concave or semi- concave [N-concave or semi-concave]. I t follows from L 10.4 t h a t an F-guide g meets no N- or S-loop in fiR. Now g meets each non-singular meridian in precisely one point [L 10.4], and since non-singular meridians are everywhere dense in R there can be no reversing point in g N R. Each element h E F * [ R is carried by a meridian k in F * [L ll.1], and k meets g. Since there is no singular point in R [L 11.2 (iii)], we conclude t h a t h E F * I R is an element in F meeting g. Hence R is the union of elements in F meeting g.
Proo/ o/ (b). Suppose an F-guide g exists without reversing points. Then B, in Th ll.1, = 0. Otherwise g would enter UN or Us and hence meet a primitive [Th 9.3], and by Th 10.2 carry at least one reversing point, contrary to hypothesis. Hence B = 0 and we infer the existence of a doubly connected meridional region [Th 11.1 (a)].
Conversely the existence of a d o u b l y connected meridional region R implies, as in the proof of (a), the existence of an F-guide without reversing point, and that R is the u n i o n of all elements in F meeting g.
The establishes Th 11.2.
There is no singular point ill a meridional region [L 11.2 (iii)], and none in a central or spiral annulus [Ths 8.1 and 8.2] or in a b o u n d a r y common to a central and a spiral an- n u l u s [Th 8.3 (b)]. Thus each singular point of F * is in Cl UN 0 C1 Us. Hence the following theorem.
Theorem 11.3. Regardless o/loop coverage or the existence of meridians, a necessary and su//icient condition that F* be non-singular is that there exist no N . or S-circuit or singular N- or S.loops.
w 12. M e r i d i a n s p r e s e n t , n o i n n e r c y c l e
When there is at least one meridian we have distinguished the case of loop coverage from the case of no loop coverage. One can equally well make a different d i v i s i o n i n t o the cases in w h i c h an inner cycle e x i s t s a n d no inner c y c l e exists.
When there is b o t h an inner cycle ~0 and a meridian, ~ is the inner c y c l e both of an N- and an S-circuit. The cycle ~ is the common curve b o u n d a r y of U~ tJ N a n d Us tJ S.
CURVE F A M I L I E S F * LOCALLY THE LF~V]~L CURVES OF A PSEUDOHARMO~/IC FUNCTION 3 9
Loop coverage t h u s occurs as i n T h 9.2 (a). I n this case Y~* is t h e ~ n i o n of p r i m i t i v e s as i n d i c a t e d i n Cor 9.2. I n this section we suppose t h a t n o i n n e r cycle exists a n d d i v i d e ~ * i n t o c a n o n i c a l p o l a r sectors.
O u r d e c o m p o s i t i o n of ~ * i n t o polar sectors is a n a l o g o u s to o u r d e c o m p o s i t i o n of F~* i n t o caps a n d a n n u l i i n w 8. W e b e g a n there w i t h a p a r t i a l ordering of i n n e r cycles, N - a n d S-cycles. W e b e g i n here with a p a r t i a l o r d e r i n g of m e r i d i a n s i n M .
Order among meridians in M. L e t 0 r e p r e s e n t t h e l o n g i t u d e of a p o i n t i n ~ * . O n M we u n d e r s t a n d t h a t t h e range a t 0 is t h e whole 0-axis. B y a parallel i n M is m e a n t a n u n e n d - i n g open arc i n M covering a parallel i n ~ * . B y t h e positive side of a m e r i d i a n x i n M we u n d e r s t a n d t h a t region i n M - x i n which 0 t a k e s on a r b i t r a r i l y large p o s i t i v e v a l u e s o n each parallel i n M. T h e negative side of x is t h e c o m p l e m e n t i n M - x of t h e p o s i t i v e side
o f X.
Two m e r i d i a n s x a n d y in M which are n o t i d e n t i c a l shall s t a n d i n t h e r e l a t i o n x < y or y > x, if y meets t h e positive b u t n o t t h e n e g a t i v e side of x, or e q u i v a l e n t l y if x m e e t s t h e n e g a t i v e b u t n o t t h e positive side of y. I f x -< y t h e set x n y m a y be e m p t y , a p o i n t , a n arc, or a half o p e n arc whose p r o j e c t i o n i n E has one e n d p o i n t i n E*, a n d a l i m i t i n g c n d p o i n t either a t N or a t S.
A I)oint p i n M has c o o r d i n a t c s [~t, O] where 0 a n d ~t arc r e s p e c t i v e l y t h e longi- t u d e a n d l a t i t u d e of p. T h e r e exists a to t) m a p p i n g T of M o n t o M such t h a t the c o o r d i n a t e s of Tp are [~, 0-t 2 ~ ] . If E is a n a r b i t r a r y set in M t h e set T " E , n = • ___2 .. . . is t e r m e d congruent to E. W e shall d e n o t e T E by E (~).
T h e covering i n M of a n N- or S - p r i m i t i v e , m c r i d i o n a l region, N- or S-loop, etc.
given in Z * , will be called b y t h e same n a m e as a s u b s e t of M. Conversely the p r o j e c t i o n i n t o Z * of v a r i o u s sets first d e f i n e d o n M , such as polar sectors, c u t sectors, etc. will be called b y t h e same n a m e as s u b s e t s of Z * .
Polar sectors. If x a n d y arc m e r i d i a n s i n M a n d if x < y g x (~), t h e i n t e r s e c t i o n of the positive side of x w i t h t h e n e g a t i v e side of y will be called a polar sector H = 1] (x, y) i n M. W h e n n o a m b i g u i t y c a n arise we speak of a polar sector as a sector. If x N y = 0, [I is connected. I f x N y is a p o i n t or arc, II has two c o m p o n e n t s , one a n N-loop i n t e r i o r , t h e o t h e r a n S-loop i n t e r i o r . I f x f i y is a half o p e n arc, 11 h a s precisely o n e c o m p o n e n t of o n e of these types. If y = x cl) t h e p r o j e c t i o n of II i n Z * has j u s t one b o u n d a r y m e r i d i a n a n d a n i n n e r closure i n Z * which is Z * .
Cut sectors. L e t II (x, y) be a sector i n M such t h a t x N y * 0, or such t h a t x N y = 0 b u t there exists a n o p e n arc ceF*MI H w i t h e n d p o i n t s i n x a n d y respectively. W e t e r m H a c u t sector. W h e n x (~ y is a p o i n t we t e r m [[ simply degenerate; w h e n x N y is a n arc or