1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. HARMONIC MAPS PARAMETRIZING ALMOST HYPERBOLIC PIECES 225

for 0 :::; j :::; k. To see this, note that analogously to (33.15) we obtain formulas of

the form

(\Jh)j (F* g) = g ( (\7^9 ,h)j dF 0 dF) + g ( dF 0 (\7^9 ,h)j dF) + J ,

where J is a polynomial expression in g and at most j - 1 covariant derivatives

of dF.

By the above, we conclude that there exists a subsequence such that F (ti) con-

verges to F-y in Ck. Since the maps F (ti ): (1-lA, hl'HJ --+ (M, g (ti)) are harmonic,

we conclude that the limit map

F-y: (1-lA, hj'HJ--+ (M,g(r))

is harmonic.

We next show that F-y is an embedding. By (33.14) and by the convergence of

F (ti) to F-y, we have

c-11x (^12) h -< j(F) -Y * x12 9(a) < - c1x12 h

for some C < oo. In particular, F-y is an immersion. On the other hand, the maps

F (ti), which limit to F-y, are embeddings (in particular, the F (ti ) are injective) for

all i. Hence, the only possibility for F-y not to be injective is that F-y (81-lA) has a

self-intersection (clearly F-Ylint('HA) is injective). However , this is impossible since

F-y (81-lA) is strictly concave (with respect to the outward normal to F-y (1-lA)).^8 We

conclude that F-y is an embedding. From the convergence in Ck we have that F-y

satisfies the CMC boundary conditions.

We now show that (33.8) holds fort=/. Since F (ti) converges to F-y in Ck,

we have that

llF;g (r) - hl'HA llck-l('HA,li) = i~~ llF (ti)* g (ti) - hl'HA llck- l('HA,h)'

which is one less derivative than we need. On the other hand, we claim that we

actually have C^00 convergence of F (ti) to F-y, so that by (33.12) we have

(33.16) llF;g (r) - hl'HA llck('HA,h) = i~~ llF (ti)* g (ti) - hl'HA llck('HA,li) :::; ~·

To see that the convergence of F (ti) to F-y is in C^00 , first note that by standard
regularity theory, F-y is actually a C^00 harmonic map because k 2'. ko 2'. 4. Since

llF;g (r) - hl'HA llck- 1 (1iA,h) :::; ~ < k ~ 1 '

where k 2'. k 0 , there exists a smooth family of C^00 harmonic embeddings

F (t) : (1-lA, hl'HJ --+ (M, g (t))

satisfying the CMC boundary conditions for t E (r - c:, "( + c:) for some c: > 0
close to isometries such that P (r) = FT By the uniqueness of these harmonic
embeddings, we have P (t) = F (t) for all t E (r - c:, "f]. Hence F (ti) = P (ti)
converges to P (r) = F-y in C^00 as claimed. This completes the proof of Claim 2.

Finally, to see the uniqueness of F (t) in (33.7), suppose that we have two

families of harmonic embeddings Fi (t) : (1-lA, hl'HJ --+ (M, g (t)), t E [a, ti], where

t 1 < t 2 , both starting at Fa and satisfying the CMC boundary conditions and

(^8) Note that, at a point of self-intersection of F-y (87-lA), if it were to exist, the two unit outward
normals are pointing in opposite directions.