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Complete manifolds with nonnegative Ricci curvature and large volume growth


Invent. math. 125, 393–404 (1996)

Complete manifolds with nonnegative Ricci curvature and large volume growth
Zhongmin Shen
Department of Mathematics, Indiana University,

Indianapolis, IN 46202-3216, USA e-mail : zshen@math.iupui.edu Oblatum 18-IX-1995

1. Introduction Let (M; g) be a complete Riemannian n-manifold with nonnegative Ricci curvature. The Bishop–Gromov theorem [BC, GLP] says that the function B(p; r )] r → vol[! is monotone decreasing, where B(p; r ) denotes the metric ball n rn around p ∈ M with radius r . De?ne M by
M

:= lim

r →∞

vol[B(p; r )] : !n r n

It is easy to show that M is independent of p ∈ M , hence it is a global geometric invariant of M . We always have
M !n r n

5 vol[B(x; r )] 5 !n r n ;

?r ? 0; ?x ∈ M

(1:1)

We say (M; g) have large volume growth if M ? 0. The purpose of this paper is to study the geometric and topological structures of complete manifolds (M; g) satisfying RicM = 0;
M

?0:

(1:2)

Complete manifolds (M; g) satisfying (1.2) (including the Ricci- at case) have been studied by several people [BKN, CT, MT, Zhu], etc. A natural question is whether manifolds satisfying (1.2) have “simple” or “?nite” topological type. Recently, G. Perelman [P1] proves that there is a small constant (n) ? 0 depending only on n such that if a complete n-manifold (M; g) with RicM = 0 satis?es M ? 1 ? (n), then M is contractible. In this paper we shall prove the following

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Theorem 1.1. Let (M; g) be a complete n-manifold satisfying (1:2). Suppose that vol[B(p; r )] 1 = M + o n?1 : (1:3) !n r n r Then (M; g) has ?nite topological type; provided that conjugate radius conjM = co ? 0; or the sectional curvature secM = Ko ? ?∞. A manifold M is said to have ?nite topological type if there is a compact domain whose boundary @ is a topological manifold such that M \ is homeomorphic to @ × [0; ∞). Abresch–Gromoll [AG] ?rst obtain a ?niteness theorem of this type for complete n-manifolds (M; g) with RicM = 0 and small diameter growth 1 diam(p; r ) = o(r n ), provided that secM = Ko ? ?∞. The sectional curvature condition can be replaced by conjM = co ? 0 by a theorem of Dai–Wei [DW], which roughly says that the angle two minimizing geodesic segments is well controlled by the bounds: RicM = (n ? 1) ; conjM = co : (1:4)

One is refered to [B1, B2] for other interesting results on manifolds satisfying (1.4). In [SW], we study complete manifolds (M; g) with RicM = 0 and small 1 volume growth vol[B(p; r )] = o(r 1+ n ). We show that such manifolds have ?nite topological type, provided that secM = Ko ? ?∞ and inf vol(B(x; 1)) = vo ? 0. By the same reason, the sectional curvature condition can be replaced by conjM = co ? 0 ([DW]). Otsu [O] constructs, for any n = 5 and ‘ = 2; : : : ; [n= 2]; a complete metric g on Rn?‘ × S‘ satisfying (1.2). Further, the metric is asymptotically nonnegatively curved. As pointed to the author by G. Perelman, using the same idea as in [P2], one may be able to construct a complete nmanifold (M; g) with RicM ? 0 and M ? 0, whose topological type is in?nite. If such a manifold exists, then either the curvature is unbounded or (1.3) is not satis?ed. The following theorem gives a topological obstruction for manifolds M admitting a complete metrics with RicM ? 0 and M ? 0. Theorem 1.2. Let (M; g) be a complete n-manifold satisfying (1:2). Suppose that RicM ? 0 outside a compact subset of M. Then M has the homotopy type of a CW complex with cells each of dimension 5 n ? 2. In particular, Hi (M; Z) = 0; i = n ? 1. The author does not know whether there is a complete n-manifold (M; g) with RicM ? 0; M ? 0, but Hn?2 (M; Z) is in?nitely generated. Without the assumption on the volume growth, Hn?2 (M; Z); n = 4, can be in?nitely generated ([SY]). These examples have small volume growth (see [SW]). See also [An] for other related results.

Complete manifolds with nonnegative Ricci curvature and large volume growth

395

The geometric structure of complete manifolds satisfying (1.2) is somewhat special. At least, we can show that the manifold is large in the sense of Gromov [G]. Theorem 1.3. Let (M; g) be a complete n-manifold satisfying (1:2). Then
x ∈M

sup vol[B(x; )] = !n

n

;

? ?0 :

(1:5)

Hence M is large in the sense of Gromov. In [G], M. Gromov de?nes a notion of largeness for complete Riemannian manifolds. Let sup vol B(M ; ) := supx∈M vol[B(x; )]. A complete Riemannian manifold (M; g) is called large campared with Rn if sup vol B(M ; ) = !n
n

;

? ?0 :

M. Gromov proves that for a complete Riemannian manifold (M; g) with secM = 0, the following conditions are equivalent. (1) M is hypersperical; (2) Fill Rad(M ) = ∞; (3) Contn?1 Rad(M ) = ∞; (4) Diamn?1 (M ) = ∞; (5) sup vol B(M ; ) = !n n ; ? ? 0. M. Cai [Ca] proves the same equivalence for complete manifolds (M; g) with RicM = 0 and injM = io . Based on the splitting theorem for singular spaces due to J. Cheeger and T.H. Colding [CC] (also compare [CG]), one can easily show that all these largeness conditions are equivalent for complete manifolds (M; g) with RicM = 0. In other words, the condition injM = io can be dropped. See further discussions in Sect. 4 below.

2. Proof of Theorem 1.1 We shall prove a more general theorem than Theorem 1.1. Let (M; g) be an n-dimensional Riemannian manifold. Let R(X; Y )Z denote the curvature tensor. For 1 5 k 5 n ? 1 and a (k + 1)-dimensional subspace V ? Tx M , de?ne the Ricci curvature RicV on V by
n

RicV (u; v) :=
i=1

g(R(u; ei )ei ; v);

u; v ∈ V ;

(k ) +1 where {ei }k i=1 is an orthonormal basis for V . We say RicM = 0 (resp. ? 0) if RicV = 0 (resp. ? 0) for all (k + 1)-dimensional subspaces V ? TM . Notice that Ric = RicTx M .

Theorem 2.1. Let (M; g) be a complete n-manifold satisfying the following
k) Ric( M = 0; M

?0 :

(2:1)

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Suppose that : kn r k +1 Then (M; g) has ?nite topological type, provided that conjugate radius conjM = co ? 0; or the sectional curvature secM = Ko ? ?∞.
M

vol[B(p; r )] = !n r n

+o

1

The main idea of the proof is to show that there is no critical points of the distance function rp (x) := d(p; x) outside a compact subset. Notice that the distance function rp is not a smooth function (on the cutlocus of p). Hence the critical points of rp are not de?ned in a usual sense. The notion of critical points of rp is introduced by Grove–Shiohama [GS]. A point q ∈ M is called a critical point of rp if for any unit vector v ∈ Tq M , there is a minimizing geodesic from q to p such that ( (0); v) 5 2 :

The isotopy lemma says that if M contains no critical points of rp outside a compact subset, then M has ?nite topological type (compare also [C]). We shall prove that under the assumption of Theorem 1.1, for su ciently large ro and x ∈ M \B(p; ro ), there is a unit vector v ∈ Tx M such that for any minimizing geodesic from x to p, ( (0); v) = 3 : 4

Thus, there is no critical points of rp outside B(p; ro ). This implies that M has ?nite topological type. Let be a closed subset of the unit tangent sphere Sp M at p ∈ M . Let B (p; r ) denote the set of points x ∈ B(p; r ) such that there is a minimizing geodesic from p to x with d dt (0) ∈ . By a standard argument [BC, GLP], one can show the following generalized Bishop –Gromov volume comparison theorem. Theorem 2.2. Let (M; g) be a complete n-manifold with RicM = 0. Let vol[B (p;r )] ? Sp M be a closed subset. Then the function r → is mono!n r n tone decreasing. For 0 ? r 5 ∞; let p (r ) denote the set of unit vectors v such that the geodesic (t ) = expp (tv) is minimizing on [0, r). Notice that
p (r2 )

?

p (r1 );

0 ? r1 ? r2 ;

p (∞)

=
r?0

p (r )

:

(2:2)

The following lemma is obvious. Lemma 2.3. Let (M; g) be complete with RicM =0 . The function r→ is monotone decreasing. vol[B
p (r )

(p; r )]

!n

rn

Complete manifolds with nonnegative Ricci curvature and large volume growth

397

Lemma 2.4. Let (M; g) be a complete n-manifold satisfying (1.2). Then vol[B
p (r ) (p; r )] = !n r n

M

?r ? 0 :

(2:3)

Proof. Observe that B(p; 2r )\B(p; r ) ? B By Theorem 2.2, we get vol[B Thus vol[B(p; 2r )] ? vol[B(p; r )] 5 vol[B
p (r ) p (r ) p (r )

(p; 2r )\B

p (r )

(p; r ):

(p; 2r )] 5 vol[B

p (r )

(p; r )]2n :

(p; 2r )] ? vol[B
p (r )

p (r )

(p; r )] (2:4)

5 (2n ? 1)vol[B

(p; r )] :

Dividing (2.4) by !n r n and letting r → ∞, one obtains (2n ? 1) Thus
r →∞ M

5 (2n ? 1) lim vol[B

vol[B

p (r )

(p; r )]

r →∞

!n

rn

:

lim

p (r ) (p; r )] = !n r n

M

:

In virtue of Lemma 2.3, one obtains (2.3). Lemma 2.5. Let (M; g) be a complete n-manifold satisfying (1:2). For x ∈ @B(p; r ); h := d(x; B Proof. Since h 5 r , B(x; h) ∪ B
p (2r ) p (2r )

(p; 2r )) 5 2

?1 n M

vol[B(p; r )] ? !n r n

1 n

M

r:

(2:5)

(p; 2r ) ? B(p; 2r ) :

(2:6)

The left hand side of (2.6) is a disjoint union. By (1.1), we have vol(B(x; h)) =
n M !n h

:

It follows from Theorem 2.2 and Lemma 2.4 2n vol[B(p; r )] = vol[B(p; 2r )] = vol(B(x; h)) + vol[B =
n M !n h
p (2r )

(p; 2r )]

+

n M !n (2r )

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Z. Shen

Thus hn = 2n This proves (2.5). To prove Theorem 2.1, we need an estimate on the excess function epq (x). Let p; q ∈ M . epq (x) is de?ned by epq (x) := d(p; x) + d(q; x) ? d(p; q) :
k) Lemma 2.6. ([AG, S]) Let (M; g) be a complete n-manifold with Ric( M = 0 for some 1 5 k 5 n ? 1. Let : [0; r ] → M be a minimizing geodesic from p to q. Then for any x ∈ M; 1 hk +1 k ; (2:7) epq (x) 5 8 s ?1 M

vol[B(p; r )] ? !n r n

M

rn :

where h = d(x; ); s = min(d(p; x); d(q; x)). We also need a lemma of Dai–Wei [DW] if we assume that conjM = co . Lemma 2.7. ([DW]) Let (M; g) be complete with RicM = ?(n ? 1) and conjM = co . There is a constant Co = C (n; co ) ? 0 such that if i : [0; ri ] → M are minimizing geodesics from p, with := max(r1 ; r2 ) 5 1 4 co ; then d( 1 (r1 ); where vi =
d i dt (0); 2 (r2 ))

5 e Co

1 2

|r1 v1 ? r2 v2 | ;

i = 1; 2.
2 : [0; 2 (0)).

Let (M; g) be as in Lemma 2.7. Let 5 1 4 co and 1 ; mizing geodesics from x to p? ; q? . Put ? := ( 1 (0); we have d(p?; q? )2 5 (2 )2 e2Co
1 2

] → M be miniBy Lemma 2.7, : (2:8)

1 ? sin2

?? 2

Rewriting (2.8) and using d(p? ; q? ) 5 2 , we obtain sin2 =1 2 sin 8 : Take Suppose that ?? 2 5 2 (e C o
1 2

? 1) +

ep? q? (x) 2
1 2

:
2

(2:9)

Let

Co = (n; co ) 5 1 4 co such that e

51+

. (2:10)

ep? q? (x) 5 2 It follows from (2.9) and (2.10) that sin2 ?? 2

2

:

5 (2 )2 = sin2

8

:

Complete manifolds with nonnegative Ricci curvature and large volume growth

399

This implies that 3 : 4 Proof of Theorem 2.1. For x ∈ M; let r = d(p; x). Let : [0; 2r ] → M be a minimizing geodesic from p to q = (2r ) such that h := d(x; ) = d(x; B p (2r ) (p; 2r )). Let 1 ; 2 be two minimizing geodesics from x to p and q, respectively. Notice that min(d(p; x); d(q; x)) = r . It follows from (2.5) and (2.7) that ?= epq (x) 5 32
+1 ? kkn M

vol[B(p; r )] ? !n r n

M

r

kn k +1

k +1 kn

:

(2:11)

We assume that conjM = co in Theorem 2.1. If follows from (2.11) that there is a ro ? 0 such that for x ∈ M \B(p; ro ); epq (x) 5 2
2

;

(2:12)

where ; are chosen as above. Let p? = 1 ( ); q? = 2 ( ). By the triangle inequality and (2.12), we get ep? q? (x) 5 epq (x) 5 2 The above argument gives 3 : 2 Thus M \B(p; ro ) contains no critical points of rp . Therefore M has ?nite topological type. If we assume that secM = Ko instead of conjM = co , then we can apply the Toponogov theorem to control the angle ? instead of Lemma 2.7. The proof is similar and simpler, so is omitted. ( 1 (0);
2 (0)) 2

:

(2:13)

=

3. Proof of Theorem 1.2 In [S] we studied the Busemann function bp , which is de?ned by bp (x) := lim {r ? d(x; @B(p; r ))} :
r →∞

One can show that the above limit always exists (M; g) is said to be proper if bp is proper for some p ∈ M , that is, the subsets, {x ∈ M; bp (x) 5 a}, are compact for all a ∈ R. Put rp (x) = d(p; x) and ep (x) = rp (x) ? bp (x). ep is called the excess function at p. We shall prove a more general theorem than Theorem 1.2. First, let us quote a theorem in [S]. Theorem 3.1. ([S]) Let (M; g) be a complete proper Riemannian n-manifold. Suppose that k) k) Ric( (3:1) Ric( M = 0; M \A ? 0

400

Z. Shen

for some compact subset A ? M . Then M has the homotopy type of a CW complex with cells each of dimension 5 k ? 1. In particular; Hi (M; Z) = 0; i = k . H. Wu [Wu] ?rst proved that the same conclusion for complete Riemannian manifolds satisfying (3.1) and secM \B = 0. Compare [Wu, Sha] for other interesting work on compact manifolds with k -convex boundary. Notice that the sectional curvature condition, secM \B = 0, implies that the manifold is proper and has ?nite topological type. Thus the sectional curvature condition here is rather strong. The properness condition is a much weak non-curvature condition, and can be applied to complete manifolds with large volume growth. Theorem 3.2. Let (M; g) be a complete Riemannian n-manifold satisfying (3:1). Suppose that M ? 0. Then M has the homotopy type of a CW complex with cells each of dimension 5 k ? 1. In particular; Hi (M; Z) = 0; i = k . To prove Theorem 3.2, it su ces to prove that the manifold in Theorem 3.2 is proper. In fact, we shall prove that ep (x)=rp (x) = 1 ? bp (x)=rp (x) approaches zero as x approaches in?nity. Thus M is proper. Let Rp denote the union of rays issuing from p. Let Rp (r ) = Rp ∩ B(p; r ). We have Rp (r ) = B p (∞) (p; r ) = expp [0; r ] p (∞) : De?ne a function hp by hp (x) = d(x; Rp ) : In [S], we have proved the following
k) Lemma 3.3. ([S]) Let (M; g) be complete with Ric( M = 0 for some 1 5 k 5 n ? 1. Then 1 hp (x)k +1 k ep (x) 5 8 : (3:2) rp (x)

Lemma 3.3 is essentially Lemma 2.6 with q at in?nity, since ep (x) = limq→∞ epq (x). We have the following estimate on ep (x)=rp (x). Lemma 3.4. Let (M; g) be a complete n-manifold satisfying (1:2). Then ep (x) 5 32 rp (x)
1 ? n? 1 M

vol[B(p; rp (x))] ? !n rp (x)n + vol[

M
1 n?1

p (rp (x ))\ p (∞)] vol(Sn?1 (1))

:

(3:3)

Proof. For the sake of simplicity, let h = hp (x); r = rp (x). Since h 5 r , B(x; h) ∪ B
p (∞)

(p; 2r ) ? B(p; 2r ) :

(3:4)

Complete manifolds with nonnegative Ricci curvature and large volume growth

401

The left hand side of (3.4) is a disjoint union. Thus vol[B(x; h)] 5 vol[B(p; 2r )] ? vol[B = vol[B(p; 2r )] ? vol[B
p (r ) p (∞)

(p; 2r )]
p (r )\ p (∞)

(p; 2r )] + vol[B

(p; 2r )] : (3:5)

It follows the Bishop–Gromov theorem that vol[B(p; 2r )] 5 2n vol[B(p; r )] : By (1.1), we have vol[B(x; h)] = It follows from (2.2) and (2.3) that
n M !n (2r ) M !n h n

(3:6)

:

(3:7)

5 vol[B

p (2r )

(p; 2r )] 5 vol[B

p (r )

(x; 2r )] :

(3:8)

By the standard argument, we have vol[B
p (r )\ p (∞)

(p; 2r )] 5 !n

vol[ p (r )\ p (∞)] (2r )n : vol(Sn?1 (1))

(3:9)

It follows from (3.5)–(3.9) that hn 5 2n
?1 M

vol[B(p; r )] ? !n r n

M

+

vol[ p (r )\ p (∞)] vol(Sn?1 (1))

rn

(3:10)

Substituting (3.10) into (3.2), one obtains (3.3). Now we are in position to prove Theorem 3.1. It follows from (2.2) that
r →∞

lim vol(

p (r )\ p (∞))

=0:

(3:11)

By (3.3) and (3.11) we have lim 1? bp (x) rp (x) = lim ep (x) =0: x→∞ rp (x )

x→∞

Thus there is a metric ball B(p; ro ) such that bp (x) =
1 2 rp (x );

?x ∈ M \B(p; ro ) :

This implies that bp is proper. Then Theorem 3.2 follows. 4. Large Riemannian manifolds The proof of Theorem 1.3 relies on the recent results of Cheeger– Colding [CC].

402

Z. Shen

Let (Mi ; pi ) be a sequence of complete Riemannian n-manifolds with RicM i = (n?1) . By the Gromov precompactness theorem [GLP], (Mi ; pi ) subconverges, in the pointed Gromov–Hausdor topology, to a complete length space (Y; y). Simply, we denote this by (Y; y) = limi (Mi ; pi ). For a metric space X , we shall denote by BX (x; r ) the metric ball with radius r around x ∈ X . Cheeger–Colding [CC] prove that if (Mi ; pi ) satisfy an additional condition (4:1) vol[BM i (pi ; 1)] = v ? 0 ; then Y has Hausdor dimension n. For all r ? 0, if xi ∈ Mi , with xi → x ∈ Y , then (4:2) vol[BM i (xi ; r )] → vol[BY (x; r )] : Further, the Bishop–Gromov inequality holds for Y (relative to the n-dimensional space form of constant curvature ). The more signi?cant result by Cheeger–Colding is the following Theorem 4.1. ([CC]) Let (Mi ; pi ) be a sequence of complete Riemannian manifolds with lim inf RicM i = 0 (uniformly on compact subsets). Let (Y; y) = limi (Mi ; pi ). Suppose that Y contains a line. Then Y splits isometrically; Y = X × R for some complete length space; X; of dimension 5 (n ? 1). Notice that if (Mi ; pi ) satisfy (4.1), then X in Theorem 4.1 has Hausdor dimension n ? 1. An outline of the proof of Theorem 1.3. Since the proof is quite standard (compare [Ca, G]), we shall only give a sketch of the proof. Let : [0; ∞) → M be an arbitrary ray. By the precompactness theorem, one may assume that (M; (ti )) converges to a length space (Y1 ; y1 ) for some ti → ∞. Notice that the minimal geodesic segment i (t ) := (ti + t ); ?ti 5 t 5 ti , converges to a line l1 in Y1 passing through y1 . It follows from Theorem 4.1 that (Y1 ; y1 ) splits isometrically, Y1 = X1 × R; y1 = (x1 ; 0). It follows from (4.2) that vol[BM ( (ri ); r )] → vol[BY (y; r )]; Therefore by (1.1), we have vol[BY1 (y; r )] =
M !n r n

?r ? 0 :

;

?r ? 0 :
X1

Since Y1 = X1 × R, there is a positive constant vol[BX1 (x1 ; r )] =
X1 !n?1 r n?1

? 0 such that ?r ? 0 :

;

For the sequence of points pi(1) := (ti ) ∈ M ,
i→∞

lim (M; pi(1) ) = (X1 × R; (x1 ; o)) :

Complete manifolds with nonnegative Ricci curvature and large volume growth

403

Assuming that we have a length space (Xk ; xk ), and a sequence of points pik ∈ M such that
i→∞

lim (M; pi(k ) ) = (Xk × Rk ; (xk ; o)) :
Xk

(4:3) ? 0 such

It follows from (1.1) and (4.2) that there a positive constant that vol[BXk (xk ; r )] = Xk !n?k r n?k ; ?r ? 0 :

In particular, Xk is non-compact. Thus there is a ray k : [0; ∞) → Xk . Since the Bishop –Gromov inequality holds in Yk := Xk × Rk ; we may assume that
i→∞

lim (Xk × Rk ; ( k (ti ); 0)) = (Yk +1 ; yk +1 ) ;

(4:4)

for some ti → ∞. The minimal geodesic segment i (t ) := ( k (t + ti ); 0); ?ti 5 t 5 ti , converges to line lk +1 ∈ Yk +1 . Clearly, Yk +1 splits isometrically, Yk +1 = Xk +1 × Rk +1 with yk +1 = (xk +1 ; 0). By (4.3) and (4.4), we can ?nd a sequence of points pik +1 ∈ M such that
i→∞

lim (M; pik +1 ) = (Xk +1 × Rk +1 ; (xk +1 ; 0)) :

Eventually, we can ?nd a sequence of points pi(n) ∈ M , such that
i→∞

lim (M; pi(n) ) = (Rn ; 0) :

By (4.2), we have vol[BM (pi(n) ; r )] → vol[Bn (0; r )] = !n r n ; This proves (1.5). Let (M; g) be a complete Riemannian manifold with RicM = 0. From Theorem 1.2, one concludes that if (M; g) is not large, then M = 0. Following the line of [Ca], we can actually prove that if sup vol B(M ; 1) 5 A!n for some A ? 1, then sup vol B(M ; ) 5 C (n; A) n?1 ; ? = 1, where C (n; A) is a constant depending only on n; A. ?r ? 0 :

References
[An] M.T. Anderson: On the topology of complete manifolds of nonnegative Ricci curvature. Topology 29 (1990) 41– 55 [AG] U. Abresch, D. Gromoll: On complete manifolds with nonnegative Ricci curvature. J. AMS 3 (1990) 355 – 374 [BC] R. Bishop, R. Crittenden: Geometry of manifolds, Academic Press, New York, 1964 [BKN] S. Bando, A. Kasue, H. Nakajima: On a construction of coordinates at in?nity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97 (1989) 313 – 349 [B1] R. Brocks: Abstandsfunktion, Riccikrummung and injektivitatsradius, Diplomarbeit, University of Munster (1993)

404 [B2]

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R. Brocks: Convexity and Ricci curvature, Comptes Rendus de l’Academie des Sciences Paris. 319, Serie I (1994) 73 – 75 [Ca] M. Cai: On Gromov’s large Riemannian manifolds. Geometriae Dedicata 50 (1994) 37– 45 [C] J. Cheeger: Critical points of distance functions and applications to geometry. Lecture Notes in Math., Springer-Verlag 1504 (1991) 1– 38 [CC] J. Cheeger, T.H. Colding: Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below. C.R. Acad. Sci. Paris, 320, 353 – 357 [CG] J. Cheeger, D. Gromoll: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Di er. Geom. 6 (1971) 119 –128 [CT] J. Cheeger, G. Tian: On the cone structure at in?nity of Ricci at manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118 (1994), 493–571 [DW] X. Dai, G. Wei: A comparison-estimate of Toponogov type for Ricci curvature. Math. Ann. 303 (1995), 297–306 [G] M. Gromov: Large Riemannian manifolds. Springer Lecture Notes in Math. 1201, 108 –121 [GLP] M. Gromov, J. Lafontaine, P. Pansu: Structures m? etrique pour les vari? et? es Riemanniennes. C? edic/Fernand, Nathan, Paris, 1981 [GS] K. Grove, K. Shiohama: A generalized sphere theorem. Ann. Math. 106 (1977) 201– 211 [MT] V.B. Marenich, V.A. Toponogov: Open manifolds of nonnegative Ricci curvature with rapidly increasing volume. Sibirsk. Mat. Zh. 26 (1995), 191–194 (Russian) [O] Y. Otsu: On manifolds of positive curvature with large diameter. Math. Z. 206 (1991) 255 – 264. [P1] G. Perelman: Manifolds of positive Ricci curvature with almost maximal volume. J. Am. Math. 7 (1994) 299 – 305 [P2] G. Perelman: Construction of manifolds of positive Ricci curvature with big volume and large betti numbers. Preprint (preliminary version) [Sha] J.P. Sha: p-convexity of manifolds with boundary. Ph.D. Thesis, State University of New York at Stony Brook, 1986 [S] Z. Shen: On Riemannian manifolds of nonnegative k th-Ricci curvature. Trans. Am. Math. Soc. 338 (1993) 289 – 310 [SW] Z. Shen, G. Wei: Volume growth and ?nite topological type. Proc. Symposia in Pure Math. 54 (1993) 539 – 549 [SY] J. Sha, D.G. Yang: Examples of manifolds of positive Ricci curvature. J. Di er. Geom. 29 (1989) 95 –103 [Wu] H. Wu: Manifolds of partially positive curvature. Indiana Univ. Math. J. 36 (1987) 525 – 548 [Zhu] S. Zhu: A ?niteness theorem for Ricci curvature in dimension three. J. Di er. Geom. 37 (1993) 711– 727


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