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# A Gap Theorem for Ends of Complete Manifolds

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## Abstract

Let (M(n), o) be a pointed open complete manifold with Ricci curvature bounded from below by -(n - 1)Lambda(2) (for Lambda greater than or equal to 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 Lambda a and the dimension n of the manifold M(n). We will give a gap theorem in this paper which shows that there exists an epsilon = epsilon(n) > 0 such that M(n) has at most two ends if Lambda a less than or equal to epsilon(n). We also give examples to show that, in dimension n greater than or equal to 4, such manifolds in general do not carry any complete metric with nonnegative Ricci Curvature for any Lambda a > 0.
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.
0002-9939/94 $1.00+$.25 per page
247
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-
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.
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,
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
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