Content uploaded by Dagang Yang

Author content

All content in this area was uploaded by Dagang Yang on Oct 20, 2014

Content may be subject to copyright.

proceedings of the

american mathematical society

Volume 123, Number 1, January 1995

A GAP THEOREM FOR ENDS OF COMPLETE MANIFOLDS

MINGLIANG CAI, TOBIAS HOLCK COLDING, AND DaGANG YANG

(Communicated by Peter Li)

Abstract. Let (Mn , o) be a pointed open complete manifold with Ricci cur-

vature bounded from below by —(n - 1)A2 (for A > 0) and nonnegative

outside the ball B(o, a). It has recently been shown that there is an upper

bound for the number of ends of such a manifold which depends only on Aa

and the dimension 71 of the manifold M" . We will give a gap theorem in this

paper which shows that there exists an e = e(n) > 0 such that M" has at

most two ends if Aa < e(n). We also give examples to show that, in dimen-

sion 7i > 4, such manifolds in general do not carry any complete metric with

nonnegative Ricci Curvature for any Aa > 0 .

1. Introduction

The Cheeger-Gromoll splitting theorem states that in a complete manifold

of nonnegative Ricci curvature, a line splits off isometrically, i.e., any nonneg-

atively Ricci curved M" is isometric to a Riemannian product Nk x Rn~k,

where yV does not contain a line (cf. [CG]). In particular, such a manifold has

at most two ends. Recently, the first-named author and independently Li and

Tarn have shown that a complete manifold with nonnegative Ricci curvature

outside a compact set has at most finitely many ends [C, LT]. At about the same

time, Liu has also given a proof of the same theorem with an additional condi-

tion that there is a lower bound on sectional curvature [L], which was removed

shortly after the appearance of [C]. In this paper, we consider manifolds with

nonnegative Ricci curvature outside a compact set and prove the following gap

theorem.

Theorem. Given n > 0, there exists an e = e(n) > 0 such that for all pointed

open complete manifolds (Mn , 0) with Ricci curvature bounded from below by

-(« -1 )A2 (for A > 0 ) and nonnegative outside the ball B(o, a), if Aa < e(n),

then Mn has at most two ends.

A natural question one would like to ask is whether this theorem can be

improved so that M" must carry a complete metric with nonnegative Ricci

curvature. Indeed, it is easy to see by volume comparison that the answer to

the above question is affirmative in dimension 2 since the Euler number of such

Received by the editors April 6, 1993.

1991 Mathematics Subject Classification. Primary 53C20.

The third author was partially supported by National Science Foundation grant DMS 90-03524.

©1994 American Mathematical Society

0002-9939/94 $1.00+ $.25 per page

247

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

248 MINGLIANG CAI, T. H. COLDING, AND DaGANG YANG

a 2-dimensional complete manifold is an upper bound for the total curvature

integral. However, such a gap theorem is the best one can have in dimensions

higher than 3 as illustrated by the following examples.

For any e > 0, by gluing two sharp cones together at the singular point, it is

easy to construct a complete metric on RxSn~2, n > 4, with Ricci curvature

bounded from below by -e and with nonnegative sectional curvature away

from a metric ball of radius 1. By applying the metric surgery techniques as in

[SY] to the manifold Sx x R x S"~2, one obtains an «-dimensional complete

manifold M of infinite homotopy type with exactly two ends and with Ricci

curvature bounded from below by -e and with nonnegative Ricci curvature

outside a metric ball of radius 1. M certainly cannot carry any complete metric

with nonnegative Ricci curvature since the Cheeger-Gromoll splitting theorem

implies that a nonnegatively Ricci curved manifold with exactly two ends must

split isometrically into the product of R with a closed manifold and therefore

has finite homotopy type.

The above examples are not valid in dimension 3 since the kind of met-

ric surgery lemmas are not available. Therefore, the following problem is of

particular interest:

Does there exist an e > 0 such that if (M, o) is a pointed noncompact

complete 3-dimensional manifold with Ricci curvature bounded from below by

-e and nonnegative outside the unit metric ball B(o, 1), then M carries a

complete metric with nonnegative Ricci curvature?

2. Proof of the theorem

There are various (but equivalent) definitions of an end of a manifold. For

the sake of our argument, we use the following (compare with [A]).

Definition 2.1. Two rays yx and y2 starting at the base point o are called cofi-

nal, if for any r > 0 and all t > r, yx(t) and y2(t) lie in the same component

of M - B(o, r). An equivalence class of cofinal rays is called an end of M.

We will denote by [y] the equivalence class of y.

Notice that the above definition does not depend on the base point o and

the particular complete metric on M. Thus the number of ends of M is a

topological invariant of M.

The following lemma is a refined version of Proposition 2.2 in [C] and can

be proved by the same argument.

Lemma 2.2. Let M be as in the theorem. If [yx] and [y2] are two different ends

of M, then for any tx, t2 >0, d(yx(tx), y2(t2)) >tx+t2-2a.

In what follows, let Mn be as in the theorem. By scaling, we may assume

that Ric(M") >-(« - 1).

Following Abresch and Gromoll in [AG], let 4>(x) be the function defined

on ß_i(o, 1) - {o}, the truncated unit ball in the hyperbolic space H", with

the following property:

A<t> = 2(n-1),

^laß_l(i) = 0-

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

A GAP THEOREM FOR ENDS OF COMPLETE MANIFOLDS 249

It is easy to see that <j>(x) = G(d(o, x)), where

Given a continuous function u: M —> R and x £ M, a continuous function

ux: M —> R is called an upper barrier of u at x if ux(x) = u(x) and u <ux .

The following lemma is a slight generalization of Theorem 2.1 in [AG].

Lemma 2.3. Let M" be a complete Riemannian manifold with Ricci curvature

bounded from below by -(« - 1). Then there exist an e = e(n) > 0 and a

S = S(n) > 0 such that

u(x) <2-2S-4e

for all x £ S(o, 1 - ô) if u: M" —> R is a continuous function which satisfies

the following properties:

(1) u(o) = 0,

(2) u>-2e,

(3) dil(w)<2,

(4) Am < 2(n - 1),

where dil(u) = supx_^ \u(x) - u(y)\/d(x, y) and the last inequality is in the

barrier sense, that is, for any x £ M and a > 0, there is an upper barrier of u

at x, uXta, such that ux<a is smooth near x and Aux,a(x) < 2(n - 1) + a.

Proof. Consider H(r) = 2r + G(r). Notice that G(l) = 0 and G'(l) = 0.

Hence H(l) = 2 and H'(r) > 0 for r close to 1, and therefore there exists a c

such that 0 < c < 1 and H(c) < 2. Now choose ô = ô(n) and e = e(n) such

that

(5) 0 < S < \ min{2 - H(c), 1 - c}

and

(6) 0<e< imin{C7(l-r5),2-//(c)-2á}.

Consider the function v(y) = u(y) - G(d(x,y)) on the annulus B(x, 1)\

B(x, c). The well-known Laplacian comparison theorem for distance functions

(cf. [EH]) implies that Av < 0 (in the barrier sense). By the maximum principle

[EH], v achieves its minimum on the boundary of the annulus. Since o is an

interior point of the domain by (5) and v(o) = u(o)-G(d(o, x)) = -G(l-ô) <

-2e by (6), there exists a point z on the boundary of the domain such that

v(z) < -2e. But on S(x, 1), v = u - G(l) = u > -2e by (2). Hence

z £ S(x, c). Combining this with (3) and (6), we conclude that

u(x) < u(z) + 2c = v(z) + H(c) < 2 - 20 - 4e.

This proves Lemma 2.3.

Remark 2.4. For a ray y in M, let by be the associated Busemann function,

i.e.,

by(x)= lim(d(y(t),x)-t).

I—>oo

It is well known (e.g., see [EH]) that, in the barrier sense, Aby < n - 1 . We are

now in position to prove the theorem.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

250

MINGLIANG CAÍ, T. H. COLDING, AND DaGANG YANG

Proof of the theorem. Let M" be as in the theorem with A = 1. Let e = e(n)

be as in Lemma 2.3. We need to show that when a < e, Mn has at most two

ends. Suppose not. Let [yx], [y2], and [73] be three different ends. Consider

u := byx + by2. We claim that u satisfies the conditions in Lemma 2.3. As

a matter of fact, (1) and (3) are clear, (4) is by Remark 2.4, and (2) is a

consequence of the triangle inequality and Lemma 2.2. From Lemma 2.3, we

conclude that

(7) u(y3(l - S)) < 2 - 20 - 4s.

On the other hand, it follows from Lemma 2.2 that for any r > 0,

"(73(0) > 2t-4a.

In particular,

u(Yi(l - ô)) >2(1 - Ô) - 4a>2 - 20 - 4e.

This clearly contradicts (7) and hence completes the proof of the theorem.

References

[A] U. Abresch, Lower curvature bounds, Toponogov's theorem and bounded topology, Ann. Sei.

École Norm. Sup. (4) 18 (1985), 651-670.

[AG] U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J.

Amer. Math. Soc. 3 (1990), 355-374.

[C] M. Cai, Ends ofRiemannian manifolds with nonnegative Ricci curvature outside a compact

set, Bull. Amer. Math. Soc. (N.S.) 24 (1991), 371-377.

[CG] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curva-

ture, J. Differential Geom. 6 (1971), 119-128.

[EH] J.-H. Eschenburg and E. Heintze, An elementary proof of the Cheeger-Gromoll splitting

theorem, Ann. Global Anal. Geom. 2 (1984), 249-260.

[L] Z. Liu, Ball covering on manifolds with nonnegative Ricci curvature near infinity, Proc.

Amer. Math. Soc. 115 (1992), 211-219.

[LT] P. Li and F. Tarn, Harmonic functions and the structure of complete manifolds, preprint,

1990.

[SY] J. P. Sha and D. G. Yang, Positive Ricci curvature on the connected sums of S" x Sm , J.

Differential Geom. 33 (1991), 127-137.

(M. Cai and T. H. Colding) Department of Mathematics, University of Pennsylvania,

Philadelphia, Pennsylvania 19104

Current address, M. Cai: Department of Mathematics and Computer Science, University of

Miami, Coral Gables, Florida 33124

E-mail address : mcaiQmath. miami. edu

Current address, T. H. Colding: Courant Institute, New York University, New York, New York

10012

(D. Yang) Department of Mathematics, Tulane University, New Orleans, Louisiana

70118