A topological rigidity theorem on open manifolds with nonnegative Ricci curvature

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A topological rigidity theorem on open manifolds with nonnegative Ricci curvature

Qiaoling Wang

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

Let M be an n-dimensional complete Riemannian manifold with nonnegative Ricci curvature. For a base point p C M we denote by B(p, r) the open geodesic ball with radius r around p and let vol[B(p, 7")] be its volume. Let c~ be the volume of the unit ball in the Euclidean space R n and define C~M by

C~M = lim vol[B(p,r)]

r-+oo c u ~ r n

It follows from the relative volume comparison t h e o r e m [BC], [Gr2] t h a t the limit at the right-hand side in the above equality exists and it does not depend on the choice of p. Thus a M is a global geometric invariant of M. T h e manifold M has large volume growth if a M > 0 . Note that, in this case,

(1.1) vol[B(p, r)] > C~Ma~,f ~ for p C M and r > 0.

T h e structure of complete n o n c o m p a c t R i e m a n n i a n manifolds with nonnegative Ricci curvature and large volume growth has received much attention. Let M be such an n-dimensional manifold. Li [L] and Anderson [A] have each proven t h a t M has finite fundamental group. Li uses the heat equation while Anderson uses volume comparison arguments to prove this theorem. Perelman [P] has shown t h a t there is a small constant s ( n ) > 0 depending only on n such t h a t if a M > l - - s ( n ) , then M is contractible. It has been shown by Cheeger and Colding [CC] t h a t the condition in Perelman's t h e o r e m actually implies t h a t M is diffeomorphic to R ~.

Shen [$2] has proven t h a t M has finite topological type, provided t h a t

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400 Qiaoling Wang

and, either the conjugate radius conjM_>C> 0 or the sectional curvature KM > K 0 >

- o c . Recall that a noncompact manifold is of finite topological type if it is homeo- morphic to the interior of a compact manifold with boundary [AG]. Generalizations of Shen's theorem have been made in [OSY], [SS] and [X1].

It should be mentioned that the first important result about topological finite- ness of a complete open manifold with nonnegative Ricci curvature is due to Ab- resch Gromoll. T h e y have proven t h a t complete open manifolds with small diameter growth, o(rl/~), nonnegative Ricci curvature and sectional curvature bounded below have finite topological type. Their theorem is proven by using an inequality referred to as the excess theorem (cf. [AG]) which has many interesting applications

(cf. [Cx], [el, [CC], [G], [OSYI, [SS], IS1], [S2], [Sol, [Xl]

and IX21).

In [Pe], Petersen proposed the following conjecture.

C o n j e c t u r e . ([Pe]) A n n-dimensional complete Riemannian manifold with nonnegative Ricci curvature and C~M > ~ is diffeomorphie to R '~. s

It has been proven in [CX] and [X2] that Petersen's conjecture is true when the sectional curvature of the manifold is bounded below and if the volume of geodesic balls around some point grows properly.

In this paper, we study complete manifolds with nonnegative Ricci curvature and large volume growth. Our purpose is to prove the following topological rigidity theorem which supports Petersen's conjecture and contains no condition on the sectional curvature of the manifolds.

T h e o r e m 1.1. Given c~e (1, 1), L)0>0 and an integer n>>_2, there exist positive constants r0=r0((~, ~o, n) and e=e((~, ~o, n) such that any complete Riemannian n- manifold M with Ricci curvature RiCM>0, C~M>Ct, conjugate radius conjM_>L)0 and

(1.2) vol[B(p,r)] c~M < rn_2+l/~ ~ C con rr~

for some p E M and all r>_ro, is diffeomorphic to R ~.

2. P r o o f o f T h e o r e m 1.1

T h r o u g h o u t the paper all geodesics are assumed to have unit speed.

Let M be an n-dimensional Riemannian manifold. For a point p E M , we denote the distance fl'om p to x by d(p, x) and set dv(x)=d(p, x). We denote by critp the criticality radius of M at p, i.e., critp is the smallest critical value for the distance function d ( p , . ) : M ~ R . The criticality radius of M is defined to be c r i t ( M ) =

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infpeM critp. We refer to [C], [Grl], [G] and [GS] for the notion of critical points of distance functions and its applications.

In [WI, Wei proved an angle version Toponogov comparison theorem for Ricci curvature for which we need some notation.

A geodesic triangle {~/0, 71,72} consists of three minimal geodesics 7~ of length L[~/i] =li which satisfy

= ( o ) ,

where the indices i and i + 1 are taken rood3.

The angle at a corner, say 7o(0), is by definition Z ( - ~ ( 1 2 ) , 7 5 ( 0 ) ) . The angle opposite to ~/~ is denoted by a~.

L e m m a 2.1. ([W]) For any s 0 > 0 and go>0 there exists a constant /3o=

s co, ~)o)>0 such that if M is an n-dimensional complete Riemannian manifold with RicM_>0 and conjugate radius conjM >-- go, and {70, 71,72} is a geodesic trian- fie contained in B(p,/3o) for some pE M , then for the geodesic triangle {~0, 71,72}

in the Euclidean plane with L[Ti]=L[7~]=li, we have

(2.1)

2 2 2

['li_l +li+ 1 - 1 i "~

a i > a ~ - s o = a r c c ~ ~ / - s ~

\ i--1 i+1 /

Actually, Wei proved a comparison estimate for manifolds with Ricci curvature bounded from below, but the above statement suffices for our purposes.

The lemma below is the key step for proving Theorem 1.1.

L e m m a 2.2. Given a E ( ~ , 1 1), ~o>0 and an integer n > 2 , there exists a pos- itive constant ro=ro(a, 6, n) such that any complete Riemannian n-manifold M with Ricci curvature RicM>0, C~M_>C~ and conjugate radius conjM>gO satisfies critp>ro, for all p E M .

Proof. Let 0o=0o(~, n)C (0, 17c) be the solution of

(2.2) ~o~176 Sinn-2 t d t = ( 1 - c t ) ~ffrsin n 2 tdt,

and set So = 1 (89 00). Let/3o =/3o(n, e0, go)---/30 (n, a, go) be as in Lemma 2.1 and

s e t

(2.3) T = cos0o cosso cos(0o+eo).

cos2(00+co)-sin 2 Co '

then T>I.

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402 Qiaoling Wang

We

claim

that

(2.4) crity > r0 ~/3~ c~176176 for y E M, 4T

which will imply the conclusion of Lemma 2.2.

Suppose on the contrary that the above

claim

is false. Then there is a point

pEM

such that critp<-r0 and thus we can find a critical point

q(r

of

dp

such that

r l :=d(p, q) <_r0.

Let

Fqp

(resp.

Fpq)

be the set of unit vectors in

TqM

(resp.

TpM)

corresponding to the set of normal minimal geodesics of M from q to p (resp. p to q). For any 0 E [0, 17c], let Fpq (0) = {u E SpMIA (u, Fpq) _< 0}. Since q is a critical point of

dp,

we have Fqp( 89 which implies t h a t Fpq contains at least two distinct vectors.

Thus there exists a constant t0>0 such that

vol(Fpq(Oo))>_v(Oo)+to,

where

v(Oo)

is the volume of a geodesic ball of radius 00 in an ( n - 1)-unit sphere. It then follows from (2.2) that vol(Fpq(00))_>(1-a)c~n l + t 0 , where c~,~ is the volume of S'~(1), the unit m-sphere.

For each

uESpM,

we denote by T(u) the distance to the cut point of p along the geodesic expp(tu), t e [0, oc).

S u b l e m m a .

For any uEFpq(Oo), we have

~-(u)<~-/30.

Pro@

We proceed by contradiction. Suppose that there is a

vEFpq(Oo)

with _~/30. Then the geodesic

7(t):=expp(tv), te

[0, 1/30], is a minimizing one.

Take a vector

wCFpq

with

Z(w, v)<00.

Let 71: [0,

r l ] ~ M

be a minimal geodesic of M from p to q with 7~(0)=w. Let z=7(1/3o), set

r2=d(q,z)

and take a minimal geodesic 72(t), tC [0, r2], from q to z. Since max{d(p,

q), d(p,

z ) } = 1/30, one can see from the triangle inequality that the geodesic triangle {7, 71,72} is contained in

B(p,/3o)-

Thus we can apply Lemma 2.1 to {7, 71,72} to get arccos (

1 F~2~_,r.~ \ 2² ² 00> Z(T'(0), 7~ (0))> 1~'0 1 @ :7'2 ~ --80 ,

- \

2-~/30rl

/1

that is,

(2.5) <

cos(eo§247

Now we use the fact that q is a critical point of dp to get a minimal geodesic 73 from q to p such t h a t / ( 7 ~ ( 0 ) , 7~(0))_< ETr.1 Applying Lemma 2.1 to the geodesic triangle {72,73, 7}, we obtain

> arccos

2rlr2

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which is equivalent to

(2.6) ~/3g < r~ +r~ + 2rlr2 sin so.

S u b s t i t u t i n g (2.5) into (2.6), one gets

(2.7) flo cos(Oo + s o ) < 4r~ + 4 r 2 sin zo < 4rl + 4 ~ / r ' ~ - ~/3orl cos(Oo + s o ) ~ / 3 g sin so.

Observe t h a t for any ~ER, we have

and (2.8)

/ ~ 2 1 ~/3o eos(0o+~o)t+ ~ g g > o,

Now consider the function f : R - ~ R defined by

f ( t ) = t + V/t 2 - 113 o cos(0o + c o ) t + ~/3o 2 sin co.

It follows from (2.8) t h a t f is strictly increasing. T h u s from (2.7) and rl _<ro, we have

/30 cos(Oo+so) < 4~.o +4@~

-

~/3o~o* eos(Oo +~o)+ ~/3g sin ~o-

B y (2.4), 4ro</3o cos(0o+So) and so we get from the above inequality t h a t (2.9) (/30 cos(0o + S o ) - 4 t o ) 2 < (16to 2 -8/3oro cos(0o + s o ) + / 3 g ) sin 2 So.

S u b s t i t u t i n g (2.4) into (2.9) and simplifying, we o b t a i n

( T - 1) ~ cos 2 (0o +~o) < (T 2 + ( 1 - 2T) sos 2 (0o +so)) sin ~ ~o,

t h a t is

(cos 2 (0o + s o ) - s i n 2 z o ) T "9 - 2 cos 2 (0o + s o ) cos 2 s o t + cos 2 (0o + so) cos 2 So < 0.

Solving this inequality, one finds

cos (0o + 2~o) cos so cos (0o + So) < T < cos 0o cos ~o cos (0o + ~o) cos ~ (0o + So) - sin 2 eo cos 2 (0o + s o ) - s i n 2 eo '

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404 Qiaoling Wang

which clearly contradicts the choice of T. This completes the proof of the sublemma. []

Now we continue on the proof of the

claim.

^Let

dV(e• (tr = ~ r d~ d ~ (r

be the volmne form in the geodesic spherical coordinates around p, where

dpp(~)

is the Riemannian measure on SpM induced by the Euclidean Lebesgue measure on

TpM

(eft [Ch, p. 112]). Since RicM >0, the Bishop-Gromov comparison theorem [BC], [Gr2] gives ~ < t ~-1 for all t > 0 . Thus, for any r_> a/30, we have from 1 the sublemma that

(2.10)

vol[B(p, r)] = ~pM d#P( ~) fomin{~(~) ~} x/g(t~, ~) dt

+ j(%M\rpq(oo) dPp(~) Lmin{T(~) r} ~ dt

pq (0o) J 0 J s ~ M - r p q (0o)

v o l [ ~ (1/~0)] @ (O~OZ~_ 1 --tO) t n-1 dt~

= vol[S(190)l + ( ~ , - ~) <,

where B ( ~ 0 ) denotes the 1/~0-ball in R%

It follows f r o m (2.10) that

lira vol[B(p, r)] _< a - - -

r ~ o o o 2 n r n <to2 n

which is a contradiction and completes the proof of our

claim. []

Before stating our next l e m m a , w e fix s o m e notation. Let M be a complete o p e n R i e m a n n i a n n-manifold. Let /~p denote the (point set) union of rays issuing f r o m p; then R p is a closed subset of M . Define a function

hp

b y

(2.11) h~(~) =d(~,R~).

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We set for r > 0

(2.12) 7 - t ( p , r ) = m a x

hp(x),

z6S(p,~.)

where S(p, r) is the geodesic sphere of radius r with center p.

Let M be an n-dimensional Riemannian manifold. If, for any point x f f M and any ( h + l ) - m u t u a l l y orthogonal unit tangent vectors e, el, ..., ekCT~M, we have

~ K(eAe~)>_O, we say t h a t the kth Ricci curvature of M is nonnegative and denote this fact by R i c ~ ) > 0 . Here, K(eAe~) denotes the sectional curvature of the plane spanned by e and ei, l < i < k .

L e m m a 2.3. Given positive numbers Co, ro and integers n > 2 and k, l < k <

n - l , there is a 5=5(00, r0, n, k ) > 0 such that any complete Riemannian n-manifold M with R i c ~ ) > 0, con.]M _> C0, critp > ro and

(2.13) 7-{(p, r) < (~r 1/(k +l)

for some p c M and all r>_ro, has infinite criticality radius at p and so is diffeomor- phic to R ~.

L e m m a 2.3 is similar to L e m m a 3.1 in [X2] where there is a lower bound on the sectional curvature instead of a lower bound on the conjugate radius of M. In the proof of L e m m a 3.1 in [X2], one can use the Toponogov comparison t h e o r e m owing to the lower bound condition on the sectional curvature. We will not include the proof of L e m m a 2.3 since it can be carried out easily by using the Toponogov type estimate of Dai Wei [DW] and modifying the arguments in [$2] and IX2].

T h e o r e m 1.1 is a special case of the following more general result.

T h e o r e m 2.4. Given c~ C (89 1), ~0>0 and integers n >_ 2 and k, l < k <_ n, there are positive constants r 0 = r 0 ( a , Co,n, k) and c = c ( a , c0,n) such that any complete Riemannian n-manifold M with R i c ~ ) _>0, aM > ~, conjugate radius conjM _> C0 and

(2.14) vol[B(p,r)] <

w~r~ - r n - ( n + k ) / ( k + l )

for some p c M and all r>_ro, has infinite criticality radius at p and thus is diffeo- morphic to R ~.

Proof. Once we have L e m m a 2.2 and L e m m a 2.3, the proof of T h e o r e m 2.4 be- comes routine. We only give an outline of it. Let r0 = r0 ((~, C0, n) be as in L e m m a 2.2;

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406 Qiaoling Wang

t h e n c r i t p > r 0 . Let a=5(oo,ro,n, k)=5(c~, g0,n, k) be as in L e m m a 2.3. We define the n u m b e r c - c ( ( ~ , a0, n, k) in T h e o r e m 2.4 as

}

(2.15) c = m i n 4 '* ; 4 ( 3 ~ 1) "

W i t h the above choice of c, by L e m m a 2.3, in order to prove T h e o r e m 2.4, we need only to show t h a t

(2.16) 7l(p, r) <_ 6r ~/(k+l) for r > r0.

T h e c o n d i t i o n (2.14) a n d t h e n o n n e g a t i v i t y of t h e Ricci c u r v a t u r e of M enable us to use t h e relative volume c o m p a r i s o n t h e o r e m a n d the same discussions as in t h e p r o o f of T h e o r e m 3.2 in [X2] to show t h a t

(2.17) hp(x) < 5r 1/(k+~) for x ~ S(p, r) a n d r > r0.

F r o m t h e definition of 7{(p, r), we know t h a t (2.16) holds. []

Acknowledgement. T h e a u t h o r would like to t h a n k t h e referee for the encour- agement a n d t h e valuable suggestions.

R e f e r e n c e s

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Received April 23, 2003 Qiaoling Wang

Departamento de Matemgtica-IE Universidade de Brasfiia

Campus Universitdrio 70910-900-Brasflia-DF Brasil

emaih wang@mat.unb.br