m a n u s c r i p t a m a t h . 2 8 , 1 5 9 - 1 8 3 ( 1 9 7 9 ) m a n u s c r i p t a m a t h e m a t i c a b y S p r i n g e f - V e r l a g 1 9 7 9

O N T H E S T R U C T U R E O F M A N I F O L D S W I T H P O S I T I V E S C A L A R C U R V A T U R E

R. S c h o e n a n d S.T. Y a u ~

D e d i c a t e d to H a n s L e w y a n d C h a r l e s B. M o r r e y , Jr.

In t h i s p a p e r , w e s t u d y t h e q u e s t i o n of w h i c h c o m p a c t m a n i f o l d s a d m i t a m e t r i c w i t h p o s i t i v e s c a l a r c u r v a t u r e . S c a l a r c u r v a t u r e is p e r h a p s the w e a k e s t i n v a r i a n t a m o n g all t h e w e l l - k n o w n i n v a r i a n t s c o n s t r u c t e d f r o m the c u r v a t u r e t e n s o r . It m e a s u r e s t h e d e v i a t i o n of t h e R i e m a n n i a n v o l u m e of the g e o d e s i c b a l l f r o m t h e e u c l i d e a n v o l u m e of t h e g e o d e s i c ball. A s a r e s u l t , it d o e s n o t t e l l us m u c h of t h e b e h a v i o r of t h e g e o d e s i c s in t h e m a n i f o l d .

T h e r e f o r e it w a s r e m a r k a b l e t h a t in 1963, L i c h n e r o w i c z [i]

w a s a b l e to p r o v e t h e t h e o r e m t h a t o n a c o m p a c t s p i n m a n i f o l d w i t h p o s i t i v e s c a l a r c u r v a t u r e , t h e r e is no h a r m o n i c spinor. A p p l y i n g

^

t h e t h e o r e m of A t i y a h - S i n g e r , it t h e n f o l l o w s t h a t t h e A - g e n u s of the m a n i f o l d is zero. L a t e r , H i t c h i n [2] f o u n d t h a t t h e v a n i s h - ing t h e o r e m of L i c h n e r o w i c z c a n a l s o b e u s e d to p r o v e t h e o t h e r K O - c h a r a c t e r i s t i c n u m b e r s d e f i n e d by M i l n o r [3] are zero.

F o r a w h i l e , it w a s n o t c l e a r w h e t h e r t h e s e a r e t h e o n l y t o p o l o g i c a l o b s t r u c t i o n s for t h e e x i s t e n c e of m e t r i c s w i t h p o s i t i v e s c a l a r c u r v a t u r e . It w a s n o t u n t i l 1977 t h a t the a u t h o r s f o u n d

R e s e a r c h s u p p o r t e d in p a r t b y NSF M C F 7 8 0 4 8 7 2 .

2 S C H O E N - Y A U

a n o t h e r t o p o l o g i c a l o b s t r u c t i o n in c o n n e c t i o n w i t h some p r o b l e m s in g e n e r a l r e l a t i v i t y . (See [4], [5], [6].) A t t h a t time, w e r e s t r i c t e d o u r a t t e n t i o n to t h r e e d i m e n s i o n a l m a n i f o l d s . W e f o u n d t h a t if t h e f u n d a m e n t a l g r o u p o f a c o m p a c t t h r e e d i m e n s i o n a l m a n i f o l d w i t h n o n - n e g a t i v e s c a l a r c u r v a t u r e c o n t a i n s a s u b g r o u p i s o m o r p h i c t o t h e f u n d a m e n t a l g r o u p of a c o m p a c t s u r f a c e w i t h g e n u s > i, t h e n

w

t h e m a n i f o l d is flat. A s s u m i n g t h e ( t o p o l o g i c a l ) c o n j e c t u r e o f W a l d h a u s e n (see [7]), o n e c a n t h e n p r o v e t h a t t h e o n l y p o s s i b l e c a n d i d a t e s for c o m p a c t t h r e e d i m e n s i o n a l o r i e n t a b l e m a n i f o l d s

w i t h n o n - n e g a t i v e s c a l a r c u r v a t u r e are f l a t m a n i f o l d s a n d m a n i f o l d s w h i c h c a n b e d e c o m p o s e d as t h e c o n n e c t e d s u m

M 1 # ... # M n # k . ( S 2 • S I)

w h e r e e a c h M i is c o v e r e d b y a h o m o t o p y s p h e r e a n d k . ( S 2 x S 1) is t h e c o n n e c t e d s u m of k c o p i e s o f S 2 • S I. It is a c o n j e c t u r e t h a t M i is in f a c t the q u o t i e n t o f the t h r e e s p h e r e b y a f i n i t e g r o u p of o r t h o g o n a l t r a n s f o r m a t i o n s . If so, t h i s w i l l g i v e a c o m - p l e t e a n s w e r to o u r q u e s t i o n for t h r e e d i m e n s i o n a l m a n i f o l d s

b e c a u s e o n e c a n p r o v e that, c o n v e r s e l y , the a b o v e m a n i f o l d s d o a d m i t m e t r i c s w i t h p o s i t i v e s c a l a r c u r v a t u r e . A t t h i s p o i n t , o n e s h o u l d m e n t i o n t h a t t h e p r o o f o f the p o s i t i v e m a s s c o n j e c t u r e [6] is m u c h m o r e d e l i c a t e t h a n the c a s e o f a c o m p a c t m a n i f o l d b e c a u s e the t o p o l o g y o f t h e s p a c e d o e s not help, a n d t h e a n a l y s i s at i n f i n i t y is m u c h m o r e i n v o l v e d .

In A u g u s t o f 1978, t h e s e c o n d a u t h o r v i s i t e d P r o f e s s o r H a w k i n g i n C a m b r i d g e w h o i n d i c a t e d t h a t a g e n e r a l i z a t i o n of the a b o v e w o r k s

to f o u r s p a c e w o u l d be of g r e a t i n t e r e s t in q u a n t u m g r a v i t y . A f t e r a m o n t h , u s i n g a n i m p o r t a n t i d e a b y the f i r s t a u t h o r , w e w e r e a b l e to s e t t l e the p o s i t i v e a c t i o n c o n j e c t u r e o f H a w k i n g . A s a n e a s y c o n s e q u e n c e of t h i s proof, w e g a i n s o m e u n d e r s t a n d i n g of c o m p a c t m a n i f o l d s w i t h p o s i t i v e s c a l a r c u r v a t u r e in h i g h e r d i m e n s i o n s .

D e f i n i n g a c l a s s o f m a n i f o l d s to b e c l a s s C, in S e c t i o n I, w e p r o v e t h a t for d i m e n s i o n ~ 7, a n y c o m p a c t m a n i f o l d w i t h p o s i t i v e

s c a l a r c u r v a t u r e m u s t b e l o n g t o c l a s s C. O n e o f the k e y f e a t u r e s that o c c u r s h e r e is t h a t if the m a n i f o l d has " e n o u g h " c o d i m e n s i o n o n e h o m o l o g y c l a s s e s to i n t e r s e c t n o n - t r i v i a l l y , t h e n it d o e s n o t a d m i t a n y m e t r i c w i t h p o s i t i v e s c a l a r c u r v a t u r e . F o r e x a m p l e , the c o n n e c t e d s u m o f a n y m a n i f o l d w i t h the t o r u s o r the s o l v a m a n i f o l d a d m i t s no m e t r i c w i t h p o s i t i v e s c a l a r c u r v a t u r e . T h e t o p o l o g i c a l c o n d i t i o n t h a t w e f i n d h e r e h a s an a n a l o g u e w i t h the c o n d i t i o n t h a t t h e s e c o n d a u t h o r [8] u s e d in s t u d y i n g t h e c i r c l e action.

W e feel t h a t the t o p o l o g i c a l c o n d i t i o n s t h a t w e find h e r e are q u i t e s a t i s f a c t o r y in l o w d i m e n s i o n s . S i n c e w e u s e t h e r e g u - l a r i t y t h e o r e m s o f m i n i m a l h y p e r s u r f a c e s , w e h a v e to r e s t r i c t o u r s e l v e s to m a n i f o l d s w i t h d i m e n s i o n < 7. T h i s is r a t h e r un- s a t i s f a c t o r y b e c a u s e one f e e l s t h a t the s i n g u l a r i t y o f the m i n i m a l h y p e r s u r f a c e s h o u l d n o t b e the o b s t r u c t i o n for the p r o o f o f t h e t h e o r e m . A f t e r w e got t h e s e r e s u l t s , P r o f e s s o r s G r o m o v a n d L a w s o n w e r e able, b y a b e a u t i f u l a r g u m e n t , t o f i n d a t o p o l o g i c a l c o n d i - t i o n w h i c h w o r k s for all d i m e n s i o n s . N a m e l y t h e y f i n d t h a t o u r c o n d i t i o n is c l o s e l y r e l a t e d to the N o v i k o v s i g n a t u r e . T h e y r e p l a c e the s i g n a t u r e o p e r a t o r in the L u s z t i g p r o o f o f the N o v i k o v

4 S C H O E N - Y A U

c o n j e c t u r e b y t h e D i r a c o p e r a t o r . A p p l y i n g the L i c h n e r o w i c z v a n i s h i n g t h e o r e m to f a m i l i e s o f o p e r a t o r s , t h e y c a n p r o v e the

^

v a n i s h i n g of a g e n e r a l i z e d A - g e n u s for s p i n m a n i f o l d s . Ex- p l o i t i n g a s i m i l a r idea, t h e y c a n a l s o d e a l w i t h a c l a s s of m a n i f o l d s w h i c h t h e y c a l l e n l a r g a b l e m a n i f o l d s . H o w e v e r , t h e y h a v e to r e s t r i c t to the c l a s s o f s p i n m a n i f o l d s . E v e n in the c a s e of s p i n m a n i f o l d s , t h e i r c o n d i t i o n d o e s n o t i n c l u d e ours.

T h e r e f o r e it w o u l d c e r t a i n l y b e of i n t e r e s t to c o m b i n e the t w o c o n d i t i o n s a n d a l s o g e n e r a l i z e our c o n d i t i o n to a r b i t r a r y d i m e n - s i o n s . C o n c e r n i n g t h i s l a s t part, w e h a v e some d e f i n i t e p r o g r e s s a n d w e h o p e to r e p o r t later.

A t t h e s a m e t i m e w e w e r e f i n d i n g the t o p o l o g i c a l c o n d i t i o n s , w e w e r e a l s o w o r k i n g o n t h e c o n s t r u c t i o n o f m a n i f o l d s w i t h positiv, s c a l a r c u r v a t u r e . A s w a s m e n t i o n e d a b o v e , in the c l a s s of t h r e e d i m e n s i o n a l m a n i f o l d s , w e c a n c o n n e c t t w o m a n i f o l d s w i t h p o s i t i v e s c a l a r c u r v a t u r e t o f o r m a n o t h e r m a n i f o l d w i t h p o s i t i v e s c a l a r c u r v a t u r e . A s t h e t h e o r y of c l a s s i f i c a t i o n of m a n i f o l d s is b a s e d o n s u r g e r i e s o n m a n i f o l d s , w e

generalize

the p r o c e d u r e for connect.

s u m to g e n e r a l s u r g e r i e s . It t u r n s o u t t h a t if o n e d o e s s u r g e r i e s w i t h c o d i m e n s i o n > 3 o n m a n i f o l d s w i t h p o s i t i v e s c a l a r c u r v a t u r e , o n e a l w a y s o b t a i n s m a n i f o l d s w i t h p o s i t i v e s c a l a r c u r v a t u r e . In fact, a m o r e g e n e r a l w a y of c o n s t r u c t i n g m a n i f o l d s w i t h p o s i t i v e s c a l a r c u r v a t u r e is p r o v i d e d i n S e c t i o n 2. W i t h t h e s e s u r g e r y r e s u l t s , w e b e l i e v e t h a t m a n i f o l d s w i t h p o s i t i v e s c a l a r c u r v a - t u r e m a y b e c l a s s i f i e d s o o n ~ e c a u s e the g e o m e t r i c p r o b l e m has

F i n a l l y w e s h o u l d m e n t i o n t h a t r e c e n t l y w e l e a r n e d t h a t P r o f e s s o r s G r o m o v a n d L a w s o n h a v e s o m e f o r m of o u r s u r g e r y r e s u l t also. W e w i s h to t h a n k S.Y. C h e n g for h i s i n t e r e s t in o u r w o r k on c o n n e c t e d s u m s s i n c e t h e t i m e w e a n n o u n c e d o u r f i r s t r e s u l t s in t h i s d i r e c t i o n at B e r k e l e y in M a r c h of 1978.

i. I n t e ~ r a b i l i t y c o n d i t i o n s for the e x i s t e n c e of a m e t r i c w i t h n o n - n e g a t i v e s c a l a r c u r v a t u r e

In t h i s s e c t i o n , w e u s e t h e t h e o r y of m i n i m a l c u r r e n t s to g i v e a t o p o l o g i c a l r e s t r i c t i o n for m a n i f o l d s to a d m i t m e t r i c s w i t h n o n - n e g a t i v e s c a l a r c u r v a t u r e .

T o see t h e p r e c i s e s t a t e m e n t of t h i s t o p o l o g i c a l r e s t r i c t i o n , w e p r o c e e d to d e f i n e i n d u c t i v e l y a c l a s s of m a n i f o l d s in t h e

f o l l o w i n g m a n n e r . L e t C 3 b e t h e c l a s s of c o m p a c t o r i e n t a b l e t h r e e d i m e n s i o n a l m a n i f o l d s M s u c h t h a t for a n y f i n i t e c o v e r i n g m a n i f o l d M of M, K I ( M ) c o n t a i n s n o s u b g r o u p w h i c h is i s o m o r p h i c t o t h e

f u n d a m e n t a l g r o u p of a c o m p a c t s u r f a c e of g e n u s ~ i. In g e n e r a l , w e s a y t h a t an n - d i m e n s i o n a l c o m p a c t o r i e n t a b l e m a n i f o l d M w i t h

^

n > 4 is of c l a s s C if e i t h e r M is s p i n a n d the A - g e n u s of M is

- - n

zero o r for a n y f i n i t e c o v e r i n g s p a c e of M, e v e r y c o d i m e n s i o n

o n e (real) h o m o l o g y c l a s s c a n be r e p r e s e n t e d b y an e m b e d d e d c o m p a c t h y p e r s u r f a c e of c l a s s C

n-l"

6 S C H O E N - Y A U

T h e o r e m i. S u p p o s e M i s a c o m p a c t o r i e n t a b l e m a n i f o l d w i t h n o n - n e g a t i v e s c a l a r c u r v a t u r e w h o s e d i m e n s i o n is n o t g r e a t e r t h a n

seven. T h e n e i t h e r M h a s z e r o R i c c i c u r v a t u r e or M is of c l a s s C n.

P r o o f . F i r s t o f all, w e o b s e r v e t h a t w e c a n a s s u m e the s c a l a r c u r v a t u r e of M is e v e r y w h e r e p o s i t i v e . O t h e r w i s e the a r g u m e n t s i n [9] s h o w t h a t t h e R i c c i c u r v a t u r e of M is i d e n t i c a l l y zero.

T h e p r o o f is d o n e b y i n d u c t i o n o n d i m e n s i o n . F o r n = 3, t h i s w a s t h e t h e o r e m p r o v e d i n [5]. F o r n ~ 4, w e p r o c e e d as f o l l o w s .

If M w e r e n o t o f c l a s s Cn, t h e n t h e L i c h n e r o w i c z v a n i s h i n g t h e o r e m

^

a n d the A t i y a h - S i n g e r i n d e x t h e o r e m s h o w t h a t the A - g e n u s of M is zero. T h e r e f o r e , b y d e f i n i t i o n of C n, f o r some f i n i t e c o v e r i n g s p a c e M o f M, s o m e c o d i m e n s i o n o n e (real) h o m o l o g y c l a s s c a n n o t b e r e p r e s e n t e d b y a n y c o m p a c t e m b e d d e d h y p e r s u r f a c e of c l a s s C

n-l"

O n t h e o t h e r hand, b y g e o m e t r i c m e a s u r e t h e o r y (see [10]), o n e c a n s h o w t h a t t h i s h o m o l o g y c l a s s c a n b e r e p r e s e n t e d b y a n o r i e n t a b l e c l o s e d e m b e d d e d h y p e r s u r f a c e H of m i n i m u m a r e a ( c o m p a r e d w i t h a l l o t h e r c l o s e d h y p e r s u r f a c e s in t h e h o m o l o g y c l a s s ) . T h e r e g u l a r i t y t h e o r y g u a r a n t e e s t h a t H is r e g u l a r if n ~ 7.

W e c l a i m t h a t H a d m i t s a m e t r i c w i t h p o s i t i v e s c a l a r c u r v a t u r e . I n d e e d , l e t R i j k s b e t h e c u r v a t u r e t e n s o r of M a n d Eij b e t h e s e c o n d f u n d a m e n t a l f o r m o f H. T h e n w e c a n c o m p u t e t h e s e c o n d v a r i a t i o n o f t h e a r e a o f H a s f o l l o w s . L e t e n b e t h e u n i t n o r m a l v e c t o r o f H

and # be an a r b i t r a r y smooth f u n c t i o n d e f i n e d on H. T h e n if we d e f o r m the h y p e r s u r f a c e H along the d i r e c t i o n #en, the second d e r i v a t i v e of the area is given by (see [ii])

(i.i) - I H (Rnn#2 % ~--- H2 i,j ij ~2) + I IV ~ 12

H

w h e r e Rnn is the Ricci c u r v a t u r e of M in the d i r e c t i o n of e n.

By the m i n i m a l i t y of H, this last q u a n t i t y m u s t be n o n - n e g a t i v e for all #. In order to make use of this fact, we use the G a u s s c u r v a t u r e f o r m u l a as follows. L e t Rijks be the c u r v a t u r e tensor of H w i t h r e s p e c t to the induced m e t r i c and el, ..., en_ 1 be a local o r t h o n o r m a l frame in H. Then the G a u s s c u r v a t u r e e q u a t i o n says

(1.2) Rijij - Rijij = Hii Hjj ij for i, j < n.

S u m m i n g (1.2), we have

'~ 2 ]i2

(1.3) ~ Rijij = . ~ Rijij + ( ~ nii) - > lj

i,j n 1,3 n i i,j

9 T h e r e f o r e , by the m i n i m a l i t y of H, the scalar c u r v a t u r e of M is (1.4)

R = . ~ - R . . . .

i,j<n 1313

= 2 ~--- R n i n i i + i ~ n Rijij

= 2 Rnn + R + ~ . 9

13 1 , 3

8 S C H O E N - Y A U

w h e r e R is the s c a l a r c u r v a t u r e of H.

T h e r e f o r e , p u t t i n g (1.4) i n t o (i.i), w e h a v e

- . . Hij) 2

I ^{Iv*l =}

H H H 1 ' 3 H

for all s m o o t h f u n c t i o n s # d e f i n e d o n H.

S i n c e R > 0 o n H, w e c o n c l u d e t h a t

-f ,f

_{H H}

f o r a l l n o n - z e r o s m o o t h f u n c t i o n s ~.

L e t A b e the L a p l a c e o p e r a t o r of H. T h e n (1.6) i m p l i e s t h a t for I ~ 0, t h e o n l y s o l u t i o n o f t h e e q u a t i o n

(n-3)

( 1 . 7 ) ~ =

is t h e z e r o f u n c t i o n .

O t h e r w i s e m u l t i p l e (1.7) b y # a n d i n t e g r a t i n g , w e h a v e

R* 2

^<

Iv, I

( i . 8 ) ~

lv, I

^{= _ 89} _ 2 A ( n - 2 )

,2 2

n - 3

H H H H

w h i c h is i m p o s s i b l e .

T h e f a c t t h a t (1.7) has n o n o n - t r i v i a l s o l u t i o n m e a n s t h a t (n-3) ~ are p o s i t i v e . It all t h e e i g e n v a l u e s of the o p e r a t o r ~ -

is w e l l - k n o w n t h a t the f i r s t e i g e n f u n c t i o n for o p e r a t o r s of t h i s f o r m c a n n o t c h a n g e sign. If u is the f i r s t (positive) e i g e n f u n c t i o n o f t h e o p e r a t o r , t h e n

(1.9) A u n - 3

4(n----~ R u = - Xu

w h e r e ~ > 0. 4

If w e m u l t i p l y t h e m e t r i c o f H b y u n-3 , t h e n t h e s c a l a r c u r v a - t u r e o f H is c h a n g e d t o

4 - n ~ 3 -i

u (~u 4 (n-2)

n - 3 A U

(Note t h a t d i m H = n - l ) . T h e r e f o r e (1.9) s h o w s t h a t H a d m i t s a m e t r i c w i t h p o s i t i v e s c a l a r c u r v a t u r e . B y t h e i n d u c t i v e h y p o t h e s i s , H is o f c l a s s C w h i c h is a c o n t r a d i c t i o n .

n - I

R e m a r k . W e f o u n d t h e a r g u m e n t of u s i n g t h e f i r s t e i g e n f u n c t i o n o f n - 3

t h e o p e r a t o r A - ~ - ~ R f r o m t h e p a p e r o f K a z d a n - W a r n e r [9].

I n o r d e r t o f i n d a c l a s s o f m a n i f o l d s w h i c h b e h a v e w e l l u n d e r

!

m a p s o f n o n - z e r o d e g r e e , w e c o n s i d e r t h e f o l l o w i n g c l a s s . L e t C 3 b e t h e c l a s s of c o m p a c t t h r e e d i m e n s i o n a l m a n i f o l d s w h i c h d o n o t a d m i t a n y n o n - z e r o d e g r e e m a p t o a c o m p a c t t h r e e d i m e n s i o n a l m a n i - f o l d M s u c h t h a t H 2 ( M ) = 0 a n d M c o n t a i n s a t w o - s i d e d i n c o m p r e s s i b l e s u r f a c e w i t h g e n u s > i. F o r n > 4, l e t C' b e t h o s e n - d i m e n s i o n a l

- - - - n

c o m p a c t m a n i f o l d s M s u c h t h a t e v e r y c o d i m e n s i o n o n e h o m o l o g y

c l a s s of M c a n b e r e p r e s e n t e d , u p t o s o m e n o n - z e r o i n t e g e r , b y a m a p f r o m a m a n i f o l d of c l a s s C' It i s c l e a r f r o m t h e s e d e f i n i t i o n s

n-l"

t h a t if M is o f c l a s s C' a n d i f t h e r e is a n o n - z e r o d e g r e e m a p f r o m n

M o n t o M' t h e n M' i s o f c l a s s C'

' n "

T h e o r e m 2. If M is a c o m p a c t n - d i m e n s i o n a l m a n i f o l d w i t h n o n - n e g a t i v e s c a l a r c u r v a t u r e a n d if n < 7, t h e n e i t h e r t h e R i c c i

m

c u r v a t u r e o f M is i d e n t i c a l l y z e r o o r M is o f c l a s s C'.

n

I0 S C H O E N - Y A U

P r o o f . T h e p r o o f is a l m o s t i d e n t i c a l to t h a t o f T h e o r e m 1 e x c e p t t h a t w e h a v e to n o t i c e t h e f o l l o w i n g r e m a r k . If M is of c l a s s C3, !

t h e n u s i n g a w e l l - k n o w n l e m m a i n t o p o l o g y (see [7] p. 62) o n e can s h o w t h a t M is o f c l a s s C 3.

c o r o l l a r y i. L e t M n b e an n - d i m e n s i o n a l c o m p a c t m a n i f o l d w i t h n ~ 7 s u c h t h a t for s o m e c o m p a c t m a n i f o l d s M i w i t h d i m e n s i o n i for 3 ! i ~ n t h e r e are m a p s f r o m M i o n t o M i _ 1 w h i c h p u l l the f u n d a m e n t a l c l a s s i n H i - l ( M i _ l ) b a c k to a n o n - t r i v i a l c l a s s in M i. S u p p o s e t h a t E2(M3) = 0 a n d M 3 c o n t a i n s a n i n c o m p r e s s i b l e s u r f a c e of g e n u s

1 w i t h t r i v i a l n o r m a l b u n d l e . T h e n M n a d m i t s n o m e t r i c w i t h n o n - n e g a t i v e s c a l a r c u r v a t u r e e x c e p t t h o s e w i t h z e r o R i c c i c u r v a t u r e .

P r o o f . T h i s f o l l o w s e a s i l y f r o m t h e d u a l i t y b e t w e e n c o h o m o l o g y a n d h o m o l o g y . T h e f o l l o w i n g c o r o l l a r y is a n e a s y c o n s e q u e n c e of

C o r o l l a r y 1.

C o r o l l a r y 2. L e t M b e a c o m p a c t m a n i f o l d w h i c h a d m i t s a n o n - z e r o d e g r e e m a p to t h e n - d i m e n s i o n a l torus. T h e n for n ! 7, the o n l y p o s s i b l e m e t r i c w i t h n o n - n e g a t i v e s c a l a r c u r v a t u r e o n M is the f l a t m e t r i c .

P r o o f . It is e a s y to c h e c k t h a t M is n o t of c l a s s C' a n d h e n c e n

t h e o n l y p o s s i b l e m e t r i c w i t h n o n - n e g a t i v e s c a l a r c u r v a t u r e o n M h a s z e r o R i c c i c u r v a t u r e . O n t h e o t h e r hand, t h e a s s u m p t i o n o n M g u a r a n t e e s t h a t t h e r e a r e n l i n e a r l y i n d e p e n d e n t h a r m o n i c o n e -

f o r m s o n M. A s M h a s z e r o R i c c i c u r v a t u r e , t h e f a m i l i a r B o c h n e r m e t h o d s h o w s t h a t t h e s e f o r m s a r e p a r a l l e l forms. It is t h e n e a s y to s h o w t h a t M is flat.

R e m a r k . I. N o t e t h a t C o r o l l a r y 2 r e m a i n s v a l i d if w e r e p l a c e t h e t o r u s b y a c o m p a c t s o l v a m a n i f o l d b e c a u s e t h e l a t t e r m a n i f o l d is c l e a r l y n o t of c l a s s C'.

n

2. In t h e a b o v e t h e o r e m s , p a r t of the c o n c l u s i o n is t h a t the R i c c i c u r v a t u r e of the m a n i f o l d is zero. T h i s is a r e s t r i c t i v e c l a s s of m a n i f o l d s as w a s s h o w n b y C h e e g e r a n d G r o m o l l [12]. T h e y p r o v e t h a t c o m p a c t m a n i f o l d s w i t h z e r o R i c c i c u r v a t u r e a r e c o v e r e d i s o m e t r i c a l l y b y t h e p r o d u c t of a e u c l i d e a n s p a c e a n d a c o m p a c t s i m p l y c o n n e c t e d m a n i f o l d w i t h z e r o R i c c i c u r v a t u r e .

2. C o n s t r u c t i o n of m e t r i c s of p o s i t i y e s c a l a r c u r v a t u r e

In t h i s s e c t i o n w e c o n s t r u c t a l a r g e c l a s s of c o m p a c t m a n i f o l d s of p o s i t i v e s c a l a r c u r v a t u r e b y s h o w i n g t h a t o n e c a n ' d o s u r g e r i e s on s u b m a n i f o l d s of c o d i m e n s i o n at l e a s t t h r e e in t h e c a t e g o r y o f p o s i t i v e s c a l a r c u r v a t u r e . L e t M b e an n - d i m e n s i o n a l c o m p a c t m a n i - fold w i t h s c a l a r c u r v a t u r e R > 0 a n d m e t r i c ds 2. L e t N b e a c o m p a c t k - d i m e n s i o n a l e m b e d d e d s u b m a n i f o l d w i t h k < n - 2. In t h e a p p e n d i x w e h a v e s h o w n the e x i s t e n c e of a p o s i t i v e f u n c t i o n u o n M ~ N s a t i s - f y i n g

12 S C H O E N - Y A U

n - 2

(2.1) Au - ~ R u = 0 o n M ~ N

~

r 2 + k - n + 0 (r 3+k-n)

u _{= i f k <} n - 3

-It 2 + k - n + 0 ( l o g r -1) if k = n-3

w h e r e r is the d i s t a n c e f u n c t i o n to N m e a s u r e d w i t h r e s p e c t to ds 2 a n d A is the L a p l a c i a n w i t h r e s p e c t to ds 2. (Here 0(f(r)) m e a n s t h a t a f t e r d i f f e r e n t i a t i n g i times, t h e f u n c t i o n is b o u n d e d by d i f / d r i 9 ) F o r a p o s i t i v e f u n c t i o n ~ on N, w e r e c a l l the f o l l o w i n

s t a n d a r d f o r m u l a f o r the s c a l a r c u r v a t u r e of the m e t r i c 4

d s ' 2 = ~n--~ ds 2 n+2

(2.2) R' = # n-2 (R# - 4(n-l) A~)

n-2

4

F o r m u l a s (2.1) a n d (2.~) s h o w t h a t the m e t r i c u n-2 ds 2 has zero s c a l a r c u r v a t u r e o n M ~ N. W e n o w let h(r) b e a s m o o t h f u n c t i o n w h i c h is z e r o o n [a, ~) for a s m a l l n u m b e r a > 0, and w e d e f i n e a m e t r i c ~-~2 = ds 2 + dh ~ dh. Thus, if w e c h o o s e c o o r d i n a t e s {x i}

n

o n M, a n d s u p p o s e ds 2 = ~ g . . d x i d x j t h e n ~ 2 = ~--- ~ i j d x i d x j

i,j= 1 13 ' i,j

w h e r e gij = gij + hxihxj" T h e C h r i s t o f f e l s y m b o l s ~.13 =

1 ~ k s - -

(gs + g s - g i j , s ) a r e t h e n e a s i l y s e e n to b e

(2.3) ~k'1] = rijk + (i + IVhl2) -I hkhij

w h e r e w e u s e the m e t r i c ds 2 to r a i s e a n d l o w e r i n d i c e s , a n d hij t h e c o v a r i a n t h e s s i a n o f h t a k e n w i t h r e s p e c t to ds 2. D i r e c t c o m p u t a t i o n t h e n s h o w s t h a t t h e s c a l a r c u r v a t u r e of ~ 2 is

R 2(I + IVht2) -I

= - RijhihJ

+ (i + IVhl2) -I g i k g j s 1 6 3 - h i s )

w h e r e ~ij = gij l+IVhl 2 is the i n v e r s e m a t r i x of gij and Rij is hih j the R i c c i tensor of ds 2. Near N, one then sees

= R + (r-2+0(r-l)) (n-k-l) (n-k-2) (h')2(l+(h,)2) -I (2.4)

+ 2(r -I + 0(i))(n-k-l) h'h"

(i+ (h') 2) 2

To c o m p u t e this formula one u s e s the fact that H(r), the H e s s i a n of r, has the f o l l o w i n g form: If X l , . . . , X k , X k + l , . . . , x n is a local c o o r d i n a t e

s y s t e m s u c h t h a t X k + 1 = . . . = x n = 0 d e f i n e s N , a n d

n

r 2 = > x~, then i=k+l

~ij ^xix~

H(r) ( x ~ i' x ~ ) = r - r 3 + 0(i) for i,j > k + 1

H(r) (~xi, ~xj) = 0(i) o t h e r w i s e .

(For the r e a d e r ' s c o n v e n i e n c e , we r e m a r k that w i t h r e s p e c t to an o r t h o n o r m a l frame field

2 2) -i

= R - 2 ( l + I V h l 2 ) - l ( ~ -- R i j h i h 9) + ((Ah) 2 - ~--- hij) (l+IVh I

- 2[(Ah)~--- h i h i j h j - ~ - hihijhjkhk] (i + IVhI2) -2

14 S C H O E N - Y A U

= R +

(Z+lVhl2) -z

!

[(n-k-l) (n-k-2)r-21h ' 12 + 2 (n-k-l) r-lh'h"

+ 0(r-l) ]h'I 2 + 0(1)h'h"] - 2(l+[Vh]2) -2 h '2

[0(r-l)h '2 + (n-k-l)r-lh'h" + 0(1)h'h"]) 4

We compute the scalar curvature R of the metric ~ 2 = un-2 ~-~2.

From (2.2) we see that

4

(2.5) R = u -n-2 (R 4(n-l) u-I Au) n-2

where barred quantities are taken with respect to ~ 2 . We see from (2.3) that

- ~

= Au -

( l + l V h l 2 ) - Z + i J h i j h k u k - " ' ' t 1 + t ~ t '~

x' ~ + j

~u

Near N we use (2.1) to get the expansion

u-l~u = u-IAu_(2+k_n) (l+iVh 12)-l(Ah) (r-l+0(1))h, + (2+k-n) (l+IVh I 2) -2hihjhi j (r-l+0 (i))h' - (l+IVhl2)-lhih j ((2+k-n)r-lrij

+ (2+k-n) (l+k-n)r-2rirj) .

Since Ah = h'Ar + h" = h'(n-k-l) (r-l+0(1))+ h" and h i = h ' ( r i + 0 ( r ) we conclude

h'h" (h')

2

u-IAu = u-IAu + (n-k-2) (r-l+0(1)) + 0 (r -1)

(i+ (h') 2) 2 i+ (h') 2

S u b s t i t u t i n g t h i s a n d (2.4) i n t o (2.5) a n d u s i n g (2.1) w e h a v e 4

R = u - n - 2 [(n-k-l) (n-k-2) ( r - 2 + 0 ( r - l ) ) ( h ' ) 2 ( l + ( h ' ) 2 ) -I (2.6)

fn(n-k-2) i) ( r - l + 0 ( 1 ) ) h ' h " ( l + ( h ' ) 2 ) -2]

- 2 ,

N o t e a l s o t h a t f o r n > k + 2 a n d n > 3, w e h a v e n ( n - k - 2 ) - 1 > 0

n - 2

E q u a t i o n (2.6) s h o w s t h a t if a is s u f f i c i e n t l y s m a l l w e h a v e Z

R > 0 p r o v i d e d h is c h o s e n so t h a t h' < 0 a n d h" > 0. If w e c h o o s e c o o r d i n a t e s x l , . . . , x k l o c a l l y o n N a n d w e c h o o s e a l o c a l o r t h o n o r m a l f r a m e ~ k + l , . . . , g n f o r t h e n o r m a l s p a c e , t h e n w e c a n d e f i n e a c o o r d i n a t e s y s t e m in a n o p e n s e t o f M as f o l l o w s : F o r e a c h y = (yk+l .... ,yn) e R n - k w i t h IYl s u f f i c i e n t l y s m a l l ,

n

s e t F ( x , y ) = e X P x ( u ~ l Y U ~ a ) ' = K t If w e t a k e (x,y) as c o o r d i n a t e s f o r M, d s 2 h a s t h e f o l l o w i n g f o r m

k

ds 2 = } ( x , y ) d x idxj + ~--- ( ~ a S + 0 ( r 2 ) ) d y ~ d y 8

i , j = l gij 5 , 8

(2.7)

+ ~ ( Y S F ~ s g ~ y + 0 ( r 2 ) ) d x i d y ~ i , ~ , B , 7

N o t e t h a t r 2 = ~ - (yU)2. L e t 6 b e c h o s e n so t h a t 0 < 6 < a a n d c h o o s e t h e c u r v e t = h(r) f o r ~ < r < a so t h a t h' < 0, h" > 0, a n d t h e c u r v e j o i n s the l i n e r = ~ i n a s m o o t h m a n n e r a t t h e p o i n t (~,to) in t h e r t - p l a n e f o r s o m e t O > 0. If w e w r i t e d s 2 i n t e r m s o f t, w e s e e t h a t r t e n d s to 6 a n d

16 S C H O E N - Y A U

d r

d r = ~ dt t e n d s to z e r o as t t e n d s to t o 9 It f o l l o w s t h a t

~ 2 = d s 2 + d h ~ d h = dt 2 + ds 2 j o i n s s m o o t h l y to the m e t r i c d t 2 + d s ~ a t t = t O w h e r e d s ~ is t h e m e t r i c i n d u c e d b y ds 2 o n t h e h y p e r s u r f a c e S~ = {r=6}.

T h u s w e m a y t a k e 4

(2.8) a-S 2 = U n-2 (dt 2 + ds~) for t ~ t o

w h i c h is a f u n c t i o n t i m e s t h e p r o d u c t m e t r i c on R x S~

W e l e t d u 2 be the m e t r i c on t h e t o t a l s p a c e of the n o r m a l b u n d l e of N g o t t e n b y l i f t i n g d s ~ v i a the i n d u c e d R i e m a n n i a n n o r m a l c o n n e c t i o n . N o t e t h a t in t e r m s of the c o o r d i n a t e s (x,y) w e h a v e

d~ 2 = ~,j gij (x'0)dxidxj + >~ (dyU) 2

+ ~ YSr~8 (x,0) 6ay

d x i d y ~ i , a , 8 , y

L e t d a ~ be t h e i n d u c e d m e t r i c on t h e n o r m a l s p h e r e b u n d l e of r a d i u s 6. F o r e > 0, w e p u l l b a c k t h e m e t r i c ds~6 f r o m Se~ to S~ v i a the m a p (x,y) § (x, cy), a n d t h e n m u l t i p l y the r a d i u s of t h e f i b e r s p h e r e b y e -2. We d e n o t e this n e w m e t r i c

F r o m (2.7) w e h a v e qe,~"

(2.9) l i m = d a ~

~+0 + q ~ , ~

in s m o o t h norm. T h u s if w e let e(t) b e a s m o o t h f u n c t i o n w h i c h c h a n g e s f r o m 1 for t ~ 2t o to 0 for t ~ 2t o + b for

~2

4

l - e ( t ) ) n'-2 (dt 2 + q e ( t ) , 6 ) f o r t < 2t ~ + b

(~ (t)u+

(2.10)

~ 2 = dt 2 + d a ~ for t ~ 2t O + b

It is a s t r a i g h t f o r w a r d c a l c u l a t i o n to s e e t h a t a m e t r i c u ~ / ( n - 2 ) ( d t 2 + q ( t ) ) h a s s c a l a r c u r v a t u r e o f t h e f o r m o f t h e f o r m

4

(2.11) utn-2(Rq+0(]ql2)+0(]~l) + u t l a q u t + o ( l ~ t l ) + o(l~r I6tl))

w h e r e d o t m e a n s d i f f e r e n t i a t i o n in t, a n d

]ql'

]q] c a n b e t a k e n as t h e m a x i m u m m a t r i x e n t r y in s o m e c o o r d i n a t e s y s t e m . It is e a s y to c h e c k t h a t f o r e a c h f i x e d e w i t h 0 < e < 1 t h e m e t r i c q e , 6 h a s s c a l a r c u r v a t u r e b o u n d e d b e l o w b y a f i x e d p o s i t i v e c o n s t a n t t i m e s 6 -2 . (This is e s s e n t i a l l y a s p e c i a l c a s e of (2.7).

S i n c e r s 6 is c o n s t a n t o n S6, it f o l l o w s f r o m (2.1) t h a t u;iAqUt_ = 0 ( 6 - i ) . It t h e n f o l l o w s f r o m (2.9), (2.10), a n d (2.11) t h a t ~Vs 2 h a s n o n - n e g a t i v e s c a l a r c u r v a t u r e if b is l a r g e a n d le], I~[ s m a l l . W e c a n n o w p r o v e t h e f o l l o w i n g t h e o r e m .

T h e o r e m 3. L e t N b e a k - d i m e n s i o n a l c o m p a c t e m b e d d e d (not n e c e s s a r i l y c o n n e c t e d ) s u b m a n i f o l d of a c o m p a c t n d i m e n s i o n a l m a n i f o l d M of p o s i t i v e s c a l a r c u r v a t u r e . S u p p o s e k < n-2.

G i v e n a n y m e t r i c o n N a n d a n y c o n n e c t i o n o n t h e n o r m a l b u n d l e v of N w e d e f i n e a m e t r i c P o n t h e t o t a l s p a c e Tv b y

u s i n g t h e c o n n e c t i o n t o l i f t t h e m e t r i c f r o m N. F o r ~ > 0 s u f f i c i e n t l y s m a l l , t h e r e is a n e i g h b o r h o o d V o f N a n d a m e t r i c Q o n M ~ V so t h a t in a n e i g h b o r h o o d o f av, Q is a

18 S C H O E N - Y A U

p r o d u c t of a l i n e w i t h t h e s p h e r e b u n d l e o f r a d i u s ~ w i t h t h e m e t r i c i n d u c e d b y P.

P r o o f . W e s i m p l y c o n n e c t t h e p r e s c r i b e d m e t r i c a n d c o n n e c t i o n t o do 2 a n d t h e R i e m a n n i a n c o n n e c t i o n w i t h o n e p a r a m e t e r

f a m i l y of m e t r i c s a n d c o n n e c t i o n s p a r a m e t r i z e d s u i t a b l y b y t.

A s a b o v e , if ~ is s u f f i c i e n t l y s m a l l , e a c h 6 - s p h e r e b u n d l e in t h e f a m i l y w i l l h a v e s c a l a r c u r v a t u r e b o u n d e d b e l o w b y a p o s i t i v e c o n s t a n t t i m e s 6 -2 . W e m a y t h e n c h o o s e t h e m e t r i c s to m o v e v e r y s l o w l y w i t h r e s p e c t to t a n d a p p l y (2.11) to f i n i s h t h e p r o o f .

In the c a s e in w h i c h the n o r m a l b u n d l e is t r i v i a l w e c a n s h o w

T h e o r e m 4. L e t M, N, k, n b e as in T h e o r e m 3. T h e r e is a n e i g h b o r h o o d V of N a n d a m e t r i c P of n o n - n e g a t i v e s c a l a r c u r v a t u r e o n M ~ V w h i c h in a n e i g h b o r h o o d of ~V is a p r o - d u c t o f a l i n e w i t h N x s n - k - l ( 6 ) , N h a v i n g a n y p r e a s s i g n e d m e t r i c a n d s n - k - l ( ~ ) b e i n g t h e s t a n d a r d s p h e r e of r a d i u s

f o r a s m a l l 6 > 0.

T h e o r e m 4 f o l l o w s b y a p p l y i n g T h e o r e m 3 w i t h a t r i v i a l c o n n e c t i o n .

C o r o l l a r y 3. T h e c o n n e c t e d s u m of t w o c o m p a c t m a n i f o l d s of p o s i t i v e s c a l a r c u r v a t u r e h a s a m e t r i c of p o s i t i v e s c a l a r c u r v a t u r e .

C o r o l l a r y 4. L e t MI, M 2 b e c o m p a c t n - d i m e n s i o n a l m a n i f o l d s of p o s i t i v e s c a l a r c u r v a t u r e a n d NI, N 2 c o m p a c t k - d i m e n s i o n a l s u b m a n i f o l d s w i t h k < n - 2. S u p p o s e t h e r e is a f i b e r p r e - s e r v i n g d i f f e o m o r p h i s m F of t h e n o r m a l b u n d l e of N 1 to t h a t of N2. T h e n e w m a n i f o l d f o r m e d b y r e m o v i n g t u b u l a r n e i g h b o r - h o o d s of N 1 a n d N 2 a n d i d e n t i f y i n g t h e b o u n d a r y s p h e r e b u n d l e s v i a F h a s a m e t r i c of p o s i t i v e s c a l a r c u r v a t u r e .

C 0 r o l l a r y 5. T h e c o n n e c t e d s u m of t w o c o n f o r m a l l y f l a t m a n i f o l d s of p o s i t i v e s c a l a r c u r v a t u r e h a s a c o n f o r m a l l y f l a t m e t r i c of p o s i t i v e s c a l a r c u r v a t u r e .

C p r o l l a r y 6. If M is a c o m p a c t m a n i f o l d o f p o s i t i v e s c a l a r c u r v a t u r e , t h e n a n y m a n i f o l d w h i c h c a n b e o b t a i n e d f r o m M b y s u r g e r i e s of c o d i m e n s i o n at l e a s t t h r e e a l s o h a s a m e t r i c of p o s i t i v e s c a l a r c u r v a t u r e .

C o r o l l a r y 7. L e t M, N b e as in T h e o r e m 3, a n d s u p p o s e N is t h e b o u n d a r y of a k + l - d i m e n s i o n a l m a n i f o l d N. S u p p o s e the n o r m a l s p h e r e b u n d l e of N e x t e n d s to a s p h e r e b u n d l e o v e r N.

T h e n the m a n i f o l d w h i c h is g o t t e n b y r e p l a c i n g a n e i g h b o r h o o d of N in M w i t h the e x t e n d e d s p h e r e b u n d l e of M h a s a m e t r i c of p o s i t i v e s c a l a r c u r v a t u r e .

R e m a r k 2. w e h a v e l e a r n e d r e c e n t l y t h a t G r o m o v a n d L a w s o n h a v e i n d e p e n d e n t l y o b t a i n e d s o m e f o r m o f o u r s u r g e r y r e s u l t s .

20 S C H O E N - Y A U

T h e p r o o f s of the c o r o l l a r i e s f o l l o w e a s i l y f r o m T h e o r e m s 3 a n d 4. C o r o l l a r y 4 f o l l o w s b y a p p l y i n g T h e o r e m 3 to MI, N 1 w i t h m e t r i c and c o n n e c t i o n g o t t e n b y p u l l i n g b a c k t h o s e f r o m N 2. N o t e t h a t b y c h o o s i n g 6 s m a l l e r if n e c e s s a r y w e c a n a l w a y s m a k e t h e r a d i i of the b o u n d a r y s p h e r e b u n d l e s of T h e o r e m 3 c o i n c i d e for N 1 and N 2. C o r o l l a r y 5 f o l l o w s b e c a u s e it is e a s y to see t h a t a m e t r i c of the f o r m

4

u ( d r 2 + r 2 d 8 2 + d h ( r ) ~ dh(r))

is c o n f o r m a l l y flat w h e r e d82 is the u n i t s p h e r e m e t r i c . To p r o v e C o r o l l a r y 6, n o t e t h a t if S k is an e m b e d d e d k- s p h e r e w i t h t r i v i a l n o r m a l b u n d l e , t h e n b y T h e o r e m 4 w e h a v e a m e t r i c of n o n - n e g a t i v e s c a l a r c u r v a t u r e on

M ~ (S k x D ~ -k) w h i c h o n the b o u n d a r y c a n be t a k e n as a p r o d u c t sk(1) x s n - k - l ( 6 ) of s t a n d a r d s p h e r e s . T a k e a m e t r i c o n D k+l w h i c h is a p r o d u c t of a line w i t h sk(1) n e a r the b o u n d a r y , a n d c h o o s e s n - k - l ( ~ ) to b e a s t a n d a r d s p h e r e of r a d i u s n, so s m a l l t h a t D k x s n - k - l ( n ) h a s p o s i t i v e s c a l a r c u r v a t u r e (note n - k - i ~ 2). S i n c e b o t h D, ~ are a r b i t r a r i l y small, w e m a y t a k e 6 = D a n d c o m p l e t e the surgery. C o r o - l l a r y 7 f o l l o w s b y a s i m i l a r a p p l i c a t i o n of T h e o r e m 3.

A p p e n d i x .

In t h i s a p p e n d i x w e s k e t c h a p r o o f of t h e e x i s t e n c e o f a p o s i t i v e f u n c t i o n u s a t i s f y i n g (2.1). W e a r e a s s u m i n g t h a t N is a c o m p a c t e m b e d d e d k - d i m e n s i o n a l s u b m a n i f o l d o f a c o m p a c t n - d i m e n s i o n a l m a n i f o l d M h a v i n g s c a l a r c u r v a t u r e R > 0. W e a r e a l s o a s s u m i n g k < n - 2. L e t G ( P , Q ) b e t h e

n - 2 G r e e n ' s f u n c t i o n o n M f o r t h e o p e r a t o r L = A - ~ R.

S i n c e R > 0, G ( P , Q ) e x i s t s , a n d b y t h e m a x i m u m p r i n c i p l e G ( P , Q ) d o e s n o t c h a n g e sign. W e t a k e G ( P , Q ) to b e p o s i t i v e .

(See [12, p. 136] f o r t h e c o n s t r u c t i o n o f G ( P , Q ) . ) G r e e n ' s f o r m u l a t h e n s a y s t h a t f o r a n y f u n c t i o n f o n M, t h e f u n c t i o n

d e f i n e d b y

f

~(P) = - | G ( P , Q ) f ( Q ) d Q M

s a t i s f i e s t h e e q u a t i o n L~ = f. W e l e t ~(P,Q) d e n o t e t h e i n t r i n s i c d i s t a n c e f r o m P to Q, a n d w e r e c a l l t h a t f o r n > 3,

(A.I) G ( P , Q ) = 0 ( ~ ( P , Q ) 2-n)

W e l e t r(P) b e t h e d i s t a n c e f r o m P to N, a n d n o t e t h a t r is a L i p s c h i t z f u n c t i o n a n d r 2 is s m o o t h i n a n e i g h b o r h o o d o f N. L e t ~o b e a s m o o t h f u n c t i o n o n M ~ N s a t i s f y i n g

~o = r 2 + k - n i n a d e l e t e d n e i g h b o r h o o d o f N. It is s t r a i g h t - f o r w a r d to c h e c k t h a t f = L ~ o s a t i s f i e s

(A.2) f = 0 ( r l+k-n) n e a r N .

22 S C H O E N - Y A U

L e t { f n } b e a s e q u e n c e o f b o u n d e d f u n c t i o n s o n N w h i c h c o n v e r g e u n i f o r m l y t o f o n c o m p a c t s u b s e t s o f M ~ N.

[

S e t ~ n ( P ) = - J G ( P , Q ) f (Q)dQ. B y (A.2) w e s e e t h a t f

M n

i s a n L 1 f u n c t i o n o n M, so t h e b o u n d e d c o n v e r g e n c e t h e o r e m i m p l i e s t h a t f o r a n y P ~ N w e h a v e ~ n ( P ) § ~(P) w h e r e

(A.3) #(P) = -I G ( P , Q ) f ( Q ) d Q

M

S t a n d a r d e l l i p t i c t h e o r y i m p l i e s t h a t ~ is s m o o t h o n M % N a n d s a t i s f i e s L # = f o n M ~ N. I t f o l l o w s t h a t t h e f u n c t i o n u = # o - # s a t i s f i e s L u = 0 o n M ~ N. W e n o w m u s t s t u d y t h e a s y m p t o t i c b e h a v i o r o f ~ n e a r N. W e w i l l b e f i n i s h e d if w e c a n s h o w

(A.4) ~ = 0 ( r 3 + k - n ) if k < n - 3

= 0 ( l o g r -1) i f k = n - 3

It w i l l t h e n f o l l o w f r o m t h e m a x i m u m p r i n c i p l e t h a t u > 0 o n M ~ N. T o c h e c k (A.4) w e t a k e a p o i n t P ~ N w i t h r(P) s m a l l , a n d l e t 0 b e t h e n e a r e s t p o i n t o f N t o P, s o t h a t r(P) = p ( 0 , P ) . W e c h o o s e a n o r m a l c o o r d i n a t e s y s t e m c e n t e r e d a t 0, c a l l i t x I .... ,x n , a n d s u p p o s e _--T' .... ~ s p a n t h e

~ x 0 x

t a n g e n t s p a c e t o N a t 0. L e t 6 b e t h e r a d i u s o f o u r n o r m a l c o o r d i n a t e b a l l w h i c h w e d e n o t e B 6 ( O ) . S i n c e w e o n l y c a r e a b o u t P c l o s e t o N, w e a s s u m e r(P) < 6/2. T h u s it f o l l o w s

from (A.I) that G(P,Q) is bounded for Q ~ B~(0), so that by (A.2) we see

(A. 5) I G(P,Q) f(Q)dQ = 0(i)

M%B~ (0)

TO show (A.4), we are left with showing

(A.6) f G(P,Q) f(Q)dQ = I 0(r3+k-n) if k < n-3 J B~ (0) 10(log r -1) if k = n-3

In our normal coordinate system, we let x I = (x l,...,x k) x 2 = (x k+l

^{, = , ,}

,x n) so that x = (Xl,X2)

^,

Using (A.I) (A.2)

^{9 ,}

and our choice of normal coordinates, we see that to prove

(A.6) it suffices to show

I ix_yl2_n ly2 II+k-n

B~ (0)

dy = 100(Ix213+k-n ) if k < n-3 (log Ix2 I-l) if k = n-3 where the integral is now a Euclidean integral in R n. This may be seen by estimating the equivalent integral

f ix_yl 2-n ly2 ll+k-n dYldY 2

Evaluating this as an iterated integral, one can first show

I ix_yi2_n

B~ (0)

dy I = 0CJx2-y2J2+k-n~

24 SCHOEN-YAU

This reduces the p r o b l e m to showing

f

Ix2-Y 212+k-n JY211+k-n dy 2 = I 0 (Ix213+k-n)

[0 (log Ix2 I-1)

if k < n-3 if k = n-3 Both of these are e l e m e n t a r y to check and we omit the proofs.

This completes the proof of existence of u satisfying (2.1).

REFERENCES

[i] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad.

Sci., Paris s4r A - B 257(1963) 7-9

[2] N. Hitchin, Harmonic spinors, A d v a n c e s in Math. 14 (1974) 1-55

[3] J. Milnor, Remarks concerning spin manifolds, Differen- tial and Combinatorial Topology, a Symposium in Honor of M a r s t o n Morse, Princeton Univ. Press, 1965, 55-62 [4] R. Schoen and S. T. Yau, Incompressible minimal sur- faces, three dimensional manifolds with nonnegative scalar curvature, and the positive mass conjecture in general relativity, Proc. Natl. Acad. Sci. 75, (6), p. 2567, 1978

[5] R. Schoen and S. T. Yau, E x i s t e n c e of incompressible minimal surfaces and the toplogy of three dimensional m a n i f o l d s w i t h n o n - n e g a t i v e scalar curvature, to appear in A n n a l s of Math.

[6] R. S c h o e n and S. T. Yau, O n the p r o o f of the p o s i t i v e m a s s c o n j e c t u r e in g e n e r a l r e l a t i v i t y , to a p p e a r in Comm. Math. Phys.

[7] J. H e m p e l , 3 - m a n i f o l d s , A n n a l s of M a t h . S t u d i e s 86, P r i n c e t o n Univ. P r e s s 1976

[8] S. T. Yau, R e m a r k s on the g r o u p of i s o m e t r i c s of a R i e m a n n i a n m a n i f o l d , T o p o l o g y 16 (1977), 2 3 9 - 2 4 7 [9] J. K a z d a n and F. W a r n e r , P r e s c r i b i n g c u r v a t u r e s ,

Proc. Symp. in P u r e M a t h . 27 (1975) 3 0 9 - 3 1 9

[I0] H. B. L a w s o n Jr., M i n i m a l v a r i e t i e s in real and c o m p l e x g e o m e t r y , Univ. of M o n t r e a l , 1974 (lecture notes)

[11]

S. S. C h e r n , M i n i m a l s u b m a n i f o l d s in a R i e m a n n i a n m a n i f o l d , Univ. of K a n s a s , 1968 (lecture notes) [12] J. C h e e g e r and D. G r o m o l l , The s p l i t t i n g t h e o r e m for

m a n i f o l d s of n o n n e g a t i v e R i c c i c u r v a t u r e , J. D i f f . Geom. 6(i), 1971

[13] G. de Rham, V a r i 4 t ~ s D i f f 4 r e n t i a b l e s , Paris, H e r m a n n , 1955

R i c h a r d S c h o e n

C o u r a n t I n s t i t u t e of M a t h e m a t i c a l S c i e n c e s 251 M e r c e r S t r e e t N e w York, N. Y. 1 0 0 1 2

S h i n g - T u n g Y a u

D e p a r t m e n t of M a t h e m a t i c s S t a n f o r d U n i v e r i i t y S t a n f o r d , C a l i f o r n i a

94305

(Received January 31, 1979)