1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. NECKS IN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE 63

in the cle-

1

l+l_topology^12 to a piece of the cylinder metric 9sn-l + du^2 on

sn-l x R More precisely, there exists an embedding

cp: B(p,c-^1 r)--+ sn-l x JR.

with cp (p) E sn-l x {O} such that on B(p, c^1 r)

lcp* (gsn-1 + du^2 ) - r-^2 hlc1e-ll+1(r- 2 h) < c.

We call cp-^1 (sn-l x {O}) the center sphere of the embedded c-neck.

Note that if cp is an embedded c-neck, for c sufficiently small, we have

the embedding

'ljJ ~ i.p-1 : sn-1 x [-c-1 + 4, c-1 - 4] --+ sn.

Later we shall abuse terminology by saying that 'l/; also is an c-neck.

REMARK 18.27. Note that if cp is an embedded c-neck, where c E

(O,c(n)], then cp is an embedded c(n)-neck. As an aside, we also note

that given any (co, ko, Lo), provided c > 0 is small enough depending on

(co, ko, Lo), an embedded c-neck as in Definition 18.26 will contain an em-

bedded (co, ko, Lo)-neck as defined in §3 of Hamilton's [94].

The following, Theorem 1.1 on p. 88 in §7 of [94] (see also Lemma 9 in

[43] by two of the authors), is what Hamilton calls 'a replacement for the

[smooth] Schoenfl.ies conjecture'.

LEMMA 18.28 (Center spheres of c (n)-necks bound smooth balls). There

exists c ( n) > 0 such that if (Nn, h) is a complete noncom pact Riemannian

manifold with positive sectional curvature and if

'l/; : sn-^1 x [-c(n)-^1 +1, c(n)-^1 - 1 J --+ sn c N

is an embedded c(n)-neck, then the center sphere 'l/; (sn-l x {O}) bounds a

smooth n-ball in N.

By the Gromoll-Meyer theorem (see [77]), Nn is diffeomorphic to JR.n.

Hence, if n = 3, then Lemma 18.28 follows from a classical result. In
particular, since a center sphere 'l/; ( S^2 x { 0}) is an embedded smooth 2-

sphere, by a result of Alexander [4], it bounds a smooth 3-ball in N.^13

However, when n = 4 the corresponding result is unknown. The smooth

4-dimensional Schoenfl.ies conjecture says the following.

CONJECTURE 18.29 (Schoenfl.ies). Every smooth embedded 3-sphere in
JR.^4 bounds a smooth 4-ball.

(^12) Here I c;- (^11) denotes the least integer greater than or equal to c:-^1.
(^13) Note that in dimension 3 the topological, piecewise-linear, and differentiable cate-
gories are the same.