- CONSTANT MEAN CURVATURE SURFACES 259
By (34.2), the linearization of 1> at (gcusp, r , 0, n - 1), with respect to the vari-
ables cp and c, is given by the continuous map
(34.5) L1> : C^2 '^0 (V; JR) x JR--+ C^0 (V; JR),
where
(34.6) LC[> (f, u) ~ D(cp,c)1>(gcusp, r, 0, n - 1) (f, u) = D.,.f + u.
The kernel of LC[> is trivial. Indeed, if D.,.f + u = 0, then fv D.,.f dC5gna t = 0 impliesthat u = 0 and D.rf = O; since fv f dC5 9 nat = 0, we have that f = 0.
Given h E C^0 (V; JR), by Hodge theory and Schauder theory, there exists a
unique f E 62 ,0. (V; JR) such that
D.rf + havg = h ,where havg denotes the average of hon (V, gflat)· Thus LC[> is onto. By the standard
Schauder estimate, we havellflb.<>(V;IR) + lhavgl :'S llhllc<>(V;IR) = llL1> (f, havg)llc<>(V;IR) ·
Thus LC[> and (L1>)-^1 are continuous bijections of Banach spaces.
STEP 3. Applying the IFT to prove the existence of CMG hypersurfaces. Givenany sufficiently small co > 0 and any r E (a+ co, b - co), by Theorem K.4, there
exists a neighborhood U of gcusp in 9J1et such that for all g EU, there exist a unique
function 'Pr E C^2 '^0 (V; (-co, co)) and a unique constant Cr E JR close ton - 1 suchthat Vr+cpr is a CMC (with respect tog) hypersurface with mean curvature equal
to -Cr. We have that Vr+cpr is near the slice { r} x V and that ('Pr, Cr) depends
smoothly on r and g (see Theorem K.3). Moreover, since r is contained in an
interval with compact closure, we may assume that U is independent of r.
STEP 4. Finishing the proof. Now suppose that g E U is C^00. Since 'Pr E
C^2 '^0 and Vr+cpr is a CMC hypersurface, we conclude that 'Pr E C^00 by applyingelliptic theory to H 9 (Vr+cpr) +Cr = 0. Thus we have obtained a family of C^00
CMC hypersurfaces {Vr+cpJrE(a+c:o,b-c:o) in [a, b] x V. Since fv 'PrdC5gnat = 0, the
hypersurfaces Vr+cp,. are distinct.
Now, for r close enough to a+co we have r+cpr :S a+2c 0 and for r close enoughto b-co we have r+cpr ~ b-2c 0. Thus, for each x EV, the function r H r+cpr (x),
r E (a+ co, b - co), takes every value in the interval [a+ 2co, b - 2co]. We conclude
that the Vr+cpr sweep out [a+ 2c 0 , b - 2c 0 ] x V. This completes the proof of the
proposition. D
REMARK 34 .2. Let A E (0, oo) be the area of one of the slices of ((a, b) x V, gcusp)·
Let g (t) be a smooth family of metrics on [a, b] x V with each metric sufficiently
close to gcusp· Let St denote the corresponding I-parameter family of C^00 CMC
hypersurfaces with area A (with respect tog (t)) given by Proposition 34.1. From
the fact that gr of the area (with respect to gcusp) of the slices {r} x V is negative,
it follows that St depends smoothly on t.
REMARK 34.3 (Is the family Vr+cpr a foliation?). One should be able to prove
that the family Vr+cpr is a foliation (in this regard, see Ye [448] for related work);
however, this is not necessary for the applications to Ricci flow.
PROBLEM 34.4. Prove that for g sufficiently close to gcusp, a neighborhood of