174 D. SALAMON, FLOER HOMOLOGYLemma 3.2. For every IE .:J(M,w) there exists a constant n = n(M,w, I) > 0
such that E( v) ;::: n for every nonconstant I-holomorphic sphere v : 5^2 --> M.Proof. Choose n > 0 such that that the following holds for every I-holomorphic
curve v : { z E C : iz - zo I < r} --> M
(24) 1 ldv(z)l^2 < n ~
lz -zol<rldv(zo)l^2 :S --;.1 ldv(z)i2.
7rT lz -zol<rThe proof, that such a constant n > 0 exists, relies on a partial differential inequality
of the form .6.e ;::: -Be^2 for the energy density e = ldvl^2. (see footnote on page 156).
Now suppose that v : 52 =CU { oo} --> M is a I-holomorphic curve with energy
E(v) < n. Then the a-priori estimate (24) holds for all r > 0, hence dv = 0, and
hence v is constant. DIf t f-+ It = It+l is a smooth family of almost complex structures compatible
with w, then n = mint n(M,w, It) > 0. Moreover, if (M,w) is monotone with
minimal Chern number N, then n = N /'r satisfies the requirements of Lemma 3.2
for every IE .:J(M,w).
Proposition 3.3 (Bubbling). Let u" E M(x-, x+; H, I) be a sequence such that
(25) sup E( u") < oo.
vThen there exist finitely many points Zj = Sj + itj E JR x 51 , j = 1, ... , £, and a
solution u of ('l) such that a subsequence of u" converges to u, uniformly with allderivatives on compact subsets of JR x 51 - { z 1 , ... , zi}. Moreover, the limit solution
satisfies
E( u) ::::; lim V->inf 00 E( u") - Rn
where n =mint n(M, w, It).
Proof. We sketch the main ideas. First, it follows from basic elliptic bootstrappingtechniques that every sequence u" E M(x-, x +; H, I) with uniformly bounded first
derivatives, i.e.(26) sup ll8su"llLoo < oo,
v
has a subsequence which converges, uniformly with all derivatives on compact sub-
sets of JR x 5^1 , to a solution u of (7). Secondly, if (26) is not satisfied then, for
every sequence (" --> z with l8su"((")I --> oo, one can prove that there exists an-other sequence z" --> z = s + it and a sequence c" --> 0 such that the rescaled
sequence v"(z) = u"(z" + E"z) has a subsequence which converges to a noncon-
st;;,nt It-holomorphic map v: C--> M.^1 The removable singularity theorem asserts
that v extends to a nonconstant Irholomorphic sphere and hence, by Lemma 3.2,(^1) The sequences zv and c:v can be found by using Hofer's lemma: Let (X, d) be a complete metric
space, f : X ---> JR be a nonnegative continuous function, x E X, and 8 > 0. Then there exists a
z E X and a positive number c: :<; 8 such that
d(x,z) < 28, sup f :<; 2f(z),
B,,( z )
c:f(z) ::::: of(x).
Apply this lemma to the function f = l8 8 uvl, the point x = (", and the constant 8
l8su"(z")l-^1 /^2 to obtain sequences z"---> z and c:" > 0 such that
c:"---> 0, c:"l8su"(z")I---> oo, sup l8su"I :<; 2l8su"(z")I.
B 0 v (z")
These sequences satisfy the assertion.