Manifolds with nonnegative Ricci curvature almost everywhere

MANIFOLDS WITH NONNEGATIVE RICeI CURVATURE ALMOST EVERYWHERE

SEONG-HUN PAENG

ABSTRACT. Under the condition of RicM ;:::-en -^l)k, injM ;::: io, we prove the existence of an€>0 such that on the region of volume

€ >0 the curvature condition of splitting theorem can be weakened.

1. Introduction

It is an important problem in Riemannian geometry to classify the complete Riemannian manifolds by curvature conditions. Splitting the- orems are concerned about the non-compact manifolds with nonnega- tive curvature. Using splitting theorem, we can prove that some finite cover of a compact Riemannian manifold with nonnegative Ricci curva- ture can be splitted toT^k xN,whereN is a compact simply connected manifold and Tk is a k-torus [3]. In [6, 8], we prove sphere theorems under weaker curvature conditions t4an the standard Ricci curvature condition RicM ~ n - 1, Le., Ricci curvature and injectivity radius bounded below and RicM ~ n - 1 on M - A where diameter ofA, diam(A) or volume of A, vol(A) are sufficiently small. Similarly to sphere theorems, we will prove a splitting theorem of compact space also holds even if the Ricci curvature conditions are not satisfied on the region of small volume.

LetMio,k be the class of n-dimensional complete Riemannian man- ifolds with RicM ~ -(n-1)kand the injectivity radius injM ~io. We obtain the following theorem;

THEOREM 1.1. Let M E M?o,k and diam(M) S d. Then there exists an^E> 0 depending only on io, n,k,d such thatifRicM ~ -(n-

Received May 9, 1998.

1991 Mathematics Subject Classification: Primary 53C20; Secondary 53C21.

Key words and phrases: Ricci curvature, splitting theorem.

Partially supported by KlAS and in part supported by BSRI and GARC- KOSEF.

1)f on M - A, vol(A) ::; E, then M is diffeomorpmc to T^k x N up to finite cover, where N is a simply connected space.

This theorem is a generalization of the following theorem due to Cai

[2].

THEOREM 1.2. Let M E M(io,n,k) and diam(M) ::; d. Then there exists an E(io,n,k,d) > 0 depending only on n,io,k,d such that

ifRicM ~

-en -

We use the compactness theorem due to Anderson and Cheeger [1]

and the rigidity result in a fixed small ball[7, 4]. The following notation will be used; if wefix81, ... ,8_{n ,}thenr(8_{1 , . . .} ,8_nlE) _~ 0andr(f) _~0 as^f -+ O.

We would like to express our gratitude to Professor Hong-JongKim for much kind and helpful advice.

2. Preliminaries

In Preliminaries, we show that the conditions of RicM ~

-en

and injM ~ io make exponential map almost isometric on a ball of fixed small radius which depends only onn,k,io.

By Broclcs' estimate on the Laplacian of the distance function, we obtain the following Jacobi field estimate on io/2-ballB(p,io/2) [4, 7, 8]. LetM be a complete Riemannian manifold with RicM ~ -(n-1)k, injM ~io and ,et) be a minimal geodesic starting from p and yet) is a Jacobi field along, such that Y(O) = 0, (Y'(O),,'(0)) = O. Let db) be the distance from p to the cut point on ,. Then io ::; db).

Define A := VVr = Hess r, so trA = c:::'r and Y' = AY. Write A(t) =^B(t)+lit. We know

iT

for some constant D(io, n,k) [4, 7]. Then we have

for some constant D = D(io, n, k) ^ifr < io/2 [4, 7, 8]. In Euc1idean space, we know that D = O. For any f >0,. we can choose a uniform

ro > 0 which is depending only on n,k,io such that Dr^1/2 < € for r < ro. Then

. {" dexp(v)11 }

1 - €~ mm Ilvll IvE TxM, x E B(p, ro)

{ Ildexp(v) 11 }

~ max II_vll /v ETxM, x E B(p,ro) ~ 1+^€

on ro-ball by the above inequality. So we can find a uniform ro > 0 such that the exponential map is almost isometric on ro-ball.

Furthermore, we know that

(2.1) -(n -l)kcothk(d(-y) - t) ~ f}.r(-y(t)) ~ (n -l)kcothkt.

For the proof, see [I). Then by the Riccati equation, we get the follow- ing estimate for r ~ d('"Y) - 8,

Then

1:/21IA,/2

~ -trA(r)+trA(io/2) +(n -l)kd(-y)

2(n - 1)

~ (n -l)kcothk(d(-y) - r)+^C(io,n,k) + . +(n -l)kd(-y) Zo

2(n- 1)

~ (n-1)kcothk8+C(io,n, k)+ . +(n -l)kd(-y) Zo

~ F(io,n, k, 8,d(-y)),

for some constant F depending only on io, n, k, 8, d(-y) since cothx is decreasing function. Then

iT

iT

i o/2 io/2

~ Vdh)F(io,n,k,8,d(-y))1/2 ⁼ H(io,n,k,8,d(-y)),

for some constant FI .

r

to/

IT

lo lo ^io/2

I

S. D(io, n, k)Ji o/2+ (IIAII+ -)

io/2 t

S. D(io,n, k)Jio/2+^F1(io,n, k, 8,d('"Y))+_to~

S.F₂(io,n,k,8,d('"'()),

for some constantF_{2 •} Then by the same method as above, we also get a uniform lower bound and upper bound of the IIYII depending only onn,k,io,8,d('"'() ont < d('"'() - 8. .

3. Estimate of the volume of the bad part

For the simplicity of argument, we only consider the sequence(Mj , 9j) inM~,ksatisfying the conditionsdiam(Mj )S.d,^RicMj 2:0 on Mj - Aj

where vol(Aj) :::; f.j and f.j --+ O. Then we know that the universal cov- ering space ofMj , Mj converges to a compact subset of CQ-Riemannian manifold X on compact subset. Shortly we use Aj instead of thelift- ing of Aj, Aj . Fix a point Pj E Mj . We may assumePj E Mj . Let '"Ye(t) = ^{expPjto and J.L} is the measure on '"Ye. We use the following notations;

dj(0) = the distance fromPj to the cut point in the direction0, where 0E sn-l C T pjMj,

~ =^{{O E sn-l C} TpjMj IJ.L('"'(e([O,dj(O)]) nAj ) < f.}, Sf,8(O) = inf{s Is> 8, 0E (~)C, JL('"Ye([8,s]) nAj )> f.}.

LEMMA 3.1. For any fixed D > 0, limj-+oo vole{exppjto I 0 E

(~)C, S;,8(O) s. ^{t <}min(dj(O), D)}) = O.

We can consider {exppjtoI^0E (~)C, Sf,8(O) s. ^{t < dj(O)} as a bad}

part for applying the Bishop-Gromov theorem. We want to show that this bad part can be ignored.

Proof Assume thatM isan element ofMf_0,k' Let{u,el,··· ,en-I}

be an orthonormal basis for the tangent space at some point q E M and let ti(t) be a Jacobi field along'"re(t) = exptO such that ti(O) =

0, ¥i'(0) =ei for i =1, .. ' ,n - l.

Define

J(u,t) = {t

o

-<n-l)(det 9(ti,l'j))! if t ~^dj(O) if t> dj(O).

Then for a region A in the unit sphere of the tangent spaceTqM,

vol{expqto Ih <t <^{t2, ()} E A}= _lA(

l

tl

Let Jj and J-kbe the J ofMj and the space form with constant curva-

. . 1 - 1

ture -k, respectively andb1(u,t) =P(u,t)n=It, b(u,t) =J-k(U,t)^n-1t.

. . . b(u, r)

In the proof of the Blshop--Gromov mequality, we see that b(u, a) ~

~«u,^{r ))' if r >} a. Simply, we use J and b instead of Jj and bi, respec- b u,a

tively. We define

C^{t2 ._} {b(U,r) I [ ]}

tl . -max b(u,s) r,s E tl,t2 .

If vol«8{)C) ^---4 0 then there are nothing to prove by the Bishop-- Gromov theorem. So we assume that limj~oovol«8{)C) > O. Let

1 / 2 ' . 1/2

Then vol(Aj ) > ^Cj vol(cPn. So vol(cPn < ^{Cj ---4} 0 as j ^---4 00 which isa contradiction. .

Now we may assume that for every direction ()E (E>~Y,

Then we know that for any E > 0, there exists a c > 8 such that

E~/2

bn-1(0,c) < ^_J_ and JL(-Yo[8,c] nA_{j )} < ^E. From this fact, we know

E

that c:::; 8!,6" Then by the above inequality, we get

(3.1)

for 81,6 < r < D. Sobn-1(0,r) < 7(8,E/€i) for r >81,6.

Let

Consequently,vol(..4j (D))(8, E) = J(e~)CJiL bn-1(0,t)dtdIJ --0 asj --

00. This completes the proof. 0

REMARK 3.2. By the above proof, we know that if Pi =

Ey4,

Pi --0 and

.lim vol({expp_to I0E (~p'_)C, 8_p^j_6(0) :::;t < min(D,dj(O))}) =O.

J--+OO 3 3 3'

This value Pj will be used in following sections.

4. Proof of Theorem

We will prove that the limit space X = ^{Rk X N} whereN contains no line. Then we can prove Theorem 1.1 by a contradiction.

Shortly, we use Ai instead of the lifting ofAi, Ai. Let 'Yi be a line inMi ^{and 'Yi(O) = Pi. Then'Yi} converge to'Yin X and we may assume that Pi --p. We follow the proof in [9].

We know that vol(Mi ) ~ vl(io,n,k) and vol(B(-Yi(ti),2ti)) :::;

v2(n, k, ti) for some VI,V2. Then the number of fundamental domains in B(-Yi(ti),2ti) is bounded byni =nl(io,n,k,ti) ^:=V2/Vl. Let

as Remark 3.2 wheretiwill be chosen below. Let 'Yo",(0) = x. Now we use the similar notation as previous section;

ei(x) ={Ox E sn-l C TxMi IJ.Lbo",([0, cf(Ox))) nAi) < Pi}, S~(Ox) =inf{s Is> a, Ox E (8ⁱ(x)r, J.Lbo",([a,sD nAi)> Pi}.

We use e^{i instead ofeibi(ti)).}

Let _~(a)be a a-tubular neighborhood of the cut locus of'Yi(ti)' We also know that the volume form b'Yi(t.)(t) has a uniform lower bound Ho(io, n, k,a,t) > 0 on~(a)Cas we see in Preliminaries.

Put D to be a bounded domain with smooth boundary in X andFi

be a diffeomorphism from D to a domain Di ⁱⁿ Mi. We consider Di as D with a metric Ft9i where9i is the metric onMi. ^Define

and

B±(x)= .!im Bt(x).

t-too

Itis an essential part of the proof of :Theorem 1.1 that .6.B+ ==0, Le., B+ is a harmonic function. Ifwe assume the almost nonnegativity of Ricci curvature, we need not check that B+ isa harmonic function [2, 10]. But in our case, we must show that B+ is harmonic.

Now we chooseti such that

P·e10(n-l)kti

t - 0

Ho(^io,n, k,a,^ti )

and ti - 00 as i - 00. It is possible to find such ti by passing to a subsequence if necessary since we know ^{ti -} O. We will show that the limit ofBt has properties as Busemann function. Let df(O) be the distance from'Yi(ti) to the cut point in the direction(J E sn-l C T'i(ti)Mi .

Then we also know that Di C B(-Yi(ti),2ti) as i ^--t 00 and the volume of bad part

converges to 0 as i --t 00 by the choice of ti, Le.,

,i

(4.1) bn- 1 (r) <(c2ti )n-1( 0 0)1/2_n~€~ <H (n k 8)p^oe2(n-1)kti

'V.(t.) _ lj - 1 _"~ ,

" • Pi

as (3.1) since the exponential growth rate for Cr^{is less than (n-1)k.}

Then

by the choice ofti, whereW_n-1 is the volume of the standard(n - 1)- sphere. Ifi is sufficiently large, b.Bt (x) has an upper bound (n - 1)cothkd(-Yi(ti),x) ~ H2(n,k) for some constant H2(n,k). Since the volume ofW_i converges to 0, we may consider the integration only on {exP'Yi(ti)(tein for computing the upper bound of liIDi-+ooID_{i b.Bt.}

Let

,9

b"+Ric(,',,')b < o.

n-1 -

For the proof, see [5}. In the case of the space of constant curvatures, the equality holds. So for Rn,

b" = o.

In this section, we only consider the direction0 E e^{i. If,9(r) E Af,}

then

(b"b - b"bl(r) = (b"b)(r) _~0,

(4.2)

(4.4)

and if"Yo(r) E Ai andr :::; di(O) then

(b"b - b"b)(r) =«b"+b)b)(r) :::; (k+l)b(r)b(r) :::; C1(io,n,k)e2(n-1)kT,

for some constant Cl depending only on io,n,k. From this, we get (b'b - b'b)(r) = iT b"b - b"b:::; C1(io,n,k)pie2(n-1)kr.

Then we have (4.3)

(

b' _ _~) r = f;b"b-b"b < piC1(io,n,k)e2(n-1)kr

b b () b(r)b(r) - b(r)b(r) ,

b' b' p·C (i n k)e2(n-1)kT n - 1 C e⁴(n-1)ktⁱp.

_ < -=- + ^t 1 0, , <2 - - +2_^{1 t - t} 0

b - b b(r)b(r) - ti ti Ho

onR;.(b)C asi - t 00 by the choice ofti and we may assume ti/2 < r <

2ti.

Combining (4.2), (4.3) with the fact (n - 1)b'^b'Yi(ti) =6Bt, we get

'Yi(ti)

[ 6B7-dV <

(1

lUi ^{t -} Ui-Ri(O) l ui nRi^{(6) t}

:::;1

<T(bl€i) +T(b),

for any fixed U cD. We can choose b> 0 arbitrarily small.

LEMMA 4.1. B+(x) 2: 1R;n fB(x,R)B+ for fixed small io/2 >

Wn -1

R>0 andx E D.

Proof. Let (r,O) be the normal coordinate system centered. at x.

The metric gi ofDi can be expressed. as gi = dr2+r2glkdOldOk, 1 :::;

l,k :::; n - 1. Let Gⁱ = det(glk)' Itis known that 6r= n -1 + 8Iog.../G.

r Br

We know that H4 S VG(r) S Hs for r S _{i o}/2 from Preliminaries.

Using this fact with (4.3) we obtain

1 ~!(Pi n 1 C (. n k) 2(n-l)kr lim u_v_v· = lim boor _

--=-

i-too & Or i-too 2 r - b(r)b(r) ,

onVi := {exptei(x) Its<f(Bx )}. Furthermore onVi,

(4.5)

~ !(Pi ~!(Pi 2t C (. k) 3(n-l)kr lim B7-_^U_V _v · < tim 2to_U_V_vv < iPi 2 to,n, e ---tO.

i-too 2 Or - i-too .2 Or - b(r)b(r)

by the choice of ti and.JGiS C₃(io,n,k)e(n-l)kr for some constants C2,C_{3 •}

Take4>~ ^E C~(R;.(c5» such that 114>~11S2 and lllnk-too4>~ ⁼ 1 in the distribution sense and 4>f= 0 on the cut locus ofri(ti),where C~(D) is the class of coo-functions with compact support inD. ThenBt_4>~

.is a smooth function. SinceBt4>f^---t Bt in distribution sense, we also have bo(Bt4>D ^{---t b.Bt} in distribution sense. By (4.4) and divergence theorem,

(4.6)

where 0< t < R. We know that

for some constant H⁶ since

/lbor - --11n-l S C(io,n,k) r

by [4]. Also we have

tivol(vtnB(x,t» S HltiPie2(n-l)kti ---t 0

(4.7)

(4.8)

from (4.1). So we obtain

lim

r

i-+oo}O }8B(x,t)nv.c ~ ~ Or

= lim

r

i-+oo}O }8B(x,t)nv,t ~ ~ & Or

Using (4.6),

lim lim f ^B7-4>~^{0 & dO}

i-+ook-+OO}8B(x,t) ~ ~ Or

~.lim_~-+ook-+oo^lim _tn~¹ _}^f_B(x,t)^6i(Bt4>f)

+ lim lim f B7"4>~^{a N dO}

i-+ook-+oo }8B(x,t) ~ ~ Or

>limlim(f +f )

- i-+ook-+oo }8B(x,t)-~ }8B(:&lt)n~

( 8(Bt_Or4>f\fGi₊^(B7"_{~ 'I'~}A~)^{ON) dO}_Or

= lim lim .!!:.- f B7"4>~v75dO.

i-+ook-+oodtJ8B(:&,t) 1. ~ Then we get

B + - ^I·· B+ - li li B+",k

Wn-l - .lillWn-l i - . m m ^{Wn-l i 'l'i}

~-+oo ~-+ook-+oo

in distribution sense and from (4.7) and (4.8), .lim lim wn - 1Bt4>f(x)

~-+ook-+oo

( 1

L

iT 1

> lim lim - B- 4>. - B· 4>- - - d t

- i-+ook-+oo tn - 1 8B(:&,T) ~ ~ ⁰ 8B(:&,t) ~ ~ Or

~ ^{Hm lim _1_}

r

i-+ook-+ootⁿ-1 }8B(x,r)

Integrating this over (0,R), we obtain that

B+(x) ~ lim Hm _1_ { B74>~^{= _1_ ( B+}

i-+ook-+ootn - ¹ J^{B(x,r) t t tn- 1} J^B(x,r)

in distribution sense. SinceB+ is continuous, we get Lemma 4.1. 0 We easily get that 6B+ _~ 0, Le., for any nonnegative Clf-function h and any8>0,

{ h6B+ = { ~hB+ ^{= .lim {} ~ihBt ^{=.lim {} h~iBt

lD lD t-+OOlDi t-+OOlDi

=_i-+oolim ( { + ( ) h~iB:r

lDi -Ri (5) lDin~(5) ^{. t} ::; 7(8)+ ^{.lim 7(8ki) = 7(8),}

t-+oo

sincei ^{--+ gin L1,p} or Ca-norm so the coefficients of Laplacian oper- ator for harmonic coordinates converges in Ca-norm [1]. Then by the same argument as (4.4), we get the above inequality. Also we know that _~iBt^--+ _~B+in distribution sense by the above argument.

By the same argument as [3, 9], we get ~(B+ +B-) == O. From

~B+, ~B- ::; 0, we obtain ~B+==0 soB+ is harmonic.

In[3],from_~B+ == 0we get that Hess(B+) = 0and"VB+isa paral- lel vector field. It is also an important step to show thatHess(B+) = 0 in our case. We will follow Cai's proof. Define b_i by the following Dirichlet problem;

~ibi=0 bilaD= B+.

Then we can prove Lemma3.1 in [2J.

LEMMA 4.2. l"Vibil2 converges, inthe strong Ll,q-topology (for1 <

q< p), to I"VB+1².

Proof. The only difference of proof occurs when we apply the Bochner- Weitzenbock formula. But we only need the integration of~il"VibiI2 so there are no obstruction to prove this lemma. 0

From this lemma, we get Hess(B+) = 0 in Lq and can prove the remainder of the proof by the same argument as [2J.

Now we obtain that there exists an€ > 0 depending only on n,k,iQ,d such that if RicM 2': 0 on M - A where vol(A) ::; ^€ then M is diffeo- morphic to T^k x N up to finite cover, where N is a compact simply connected space. But the complete proof of Theorem 1.1 is the same as the above argument.

References

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[3] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature,J. Diff. Geom. 6 (1971), 119-128.

[4] X. Dai, Z. Shen and G. Wei, Negative Ricci curvature and isometry group, Duke Math. J. 76 (1994), 59-73.

[5] S. Gallot, D. Hullin andJ. Lafontain, Riemannian geometry, Springer-Verlag, 1980.

[6] S.-H. Paeng, A sphere theorem under a curvature perturbation, Kyushu J.

Math. 50 (1996), 459-470.

[7] , Topological entropy for geodesic flows under a Ricci curvature condi- tion, Proc. Amer. Math. Soc. 125 (199'7), 1873-1879.

[8] , A sphere theorem under a curvature perturbation11, Kyushu J. Math.

52 (1998), 439-454.

[9] R. Schoen and S. -T. Yau, Lectures on Differential Geometry, International Press, 1994.

[10] Z. Shen, G. Wei, On Riemannian manifolds of almost nonnegative curvature, Indiana Univ. Math. J. 40 (1991), 551-565.

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