260 34. CMC SURFACES AND HARMONIC MAPS BY IFT
1.2. Existence of CMC spheres in necks.
Analogous to Proposition 33.11 we have the following consequence of the IFT
for necks. This result may be of interest for studying neck formation in Ricci flow.
PROPOSITION 34.5 (Existence of CMC spheres in necks). Given [a, b] CIR and
E: > 0, if a c= metric g on [a, b] x s n-l is sufficiently close in the C^2 '°'-topology
to the standard cylinder metric 9cyl = dr^2 + 9sphi where ( sn-i, 9sph) is the unit
sphere, then there exists a smooth 1-parameter family of CMG (with respect tog)
hypersurfaces which sweep out [a+ E:, b - c] x sn-l and which are C^2 '°'-close to the
standard slices.
PROOF. The proof is essentially exactly the same as in Proposition 34.1, using
Theorem K.4, except for the following:
(1) (vn-^1 ,gflat) is replaced by (sn-^1 ,gsph)·
(2) The hyperbolic cusp metric 9cusp = dr^2 + e-^2 r 9flat is replaced by the
standard cylinder metric 9cyl = dr^2 + 9sph.
(3) We have for 9cyl that(34.7) Re (:r' :r) = 0 and IIIl^2 = 0
for the slices { r} x sn-l.
Let Sep~ { (cp (x), x): x E 5n-l} and define
(g, r, cp, c) ~ H 9 (Sr+cp) + c
analogously to (34.4). Note that <P(gcyli r, 0, 0) = 0. Now using (34.1), we find that
just as in (34.6), we have
D(cp,c) <P(gcyli r , 0 , 0) (!, u) = b..f + u.
There are no other essential differences in the proof; we leave it to the reader to
verify this. D
2. Harmonic maps near the identity of sn
Isometries of Riemannian manifolds are harmonic maps. However, for the unit
sphere, there are other harmonic self-maps. For example, Smith [387] proved that
for n ::::; 7 there exists a harmonic map of s n to itself in every homotopy class. For
maps near the identity, we have the following.
PROPOSITION 34.6 (Harmonic maps of the unit 5n near id). For n ~ 3 aharmonic map of the unit n-sphere sn to itself which is sufficiently close to the
identity in the C^2 '°'-norm must be an isometry. A harmonic map of the round 2-
sphere to itself which is sufficiently close to the identity in the C^2 '°' -norm must be
a conj ormal diffeomorphism.In the rest of this section we shall prove this result using the IFT. Note thatwhen n = 2, this is a classical result since any nonconstant harmonic map from
a 2-sphere into a Riemannian manifold of any dimension at least 2 is a conformal
map (in fact, such a map is a branched conformal minimal immersion by Corollary
1.7 in Sacks and Uhlenbeck [342]).