LECTURE 4. GROMOV COMPACTNESS AND STABLE MAPS 191This implies that there can be only finitely many points z 1 , ... , ze near which the
derivative of uv tends to infinity. After passing to a suitable subsequence, we
may assume, by Remark 4.1 (iii), that uv converges uniformly with all derivatives
on compact subset of <C - {z1, ... , ze} to a J-holomorphic curve u. Invoking the
removable singularity theorem, we find that u extends to a J-holomorphic curve on
S^2 = <C u { 00}.
These arguments exhibit the convergence behaviour of a sequence uv of J-ho-
lomorphic spheres with uniformly bounded energy. They do not, however, give
a complete picture of the bubble tree. At each point, near which the derivatives
of uv blow up, several J-holomorphic spheres may bubble off and, moreover, the
collection of all these spheres forms a connected set. To see this it is necessary to
refine the above rescaling argument.
Soft rescaling
Let us assume that uv : <C __, M is a sequence of J-holomorphic curves with
uniformly bounded energy E(uv) :::; c and that a subsequenc has been chosen (still
denoted by uv) which converges, uniformly with all derivatives on compact subsets
of <C-{z 1 , ... , ze}, to a J-holomorphic curve u: <C __, M. Suppose further that the
function z r-t lduv(z)I attains its maximum in the (small) ball Br(z 1 ) at the point
zj __, z 1 and that this maximum tends to oo. Then the above discussion shows that
m(z1) = lim lim E(uv, B E(z1)) 2: Ii
e---tO v---toofor every j. Here we assume that a subsequence has been chosen such that the
relevant limits exist. We shall see that this number m( z 1 ) is equal to the total
energy of all the bubbles splitting off at the point z 1. To capture the ''first" J-
holomorphic sphere bubbling off at zj we choose c'j > 0 such that
(37) E(uv, B 0 j (zj)) = m(z1) - ~-
By definition of m(z 1 ), the sequence c'j > 0 converges to zero. Now consider the
sequence
vj(z) = uv(zj + c'jz).
A suitable subsequence, still denoted by vj, has the following properties (see Fig-
ure 12).
(a) The function z r-t ldvj(z)I takes on its supremum (over a large ball) at 0:
ldv'j(O)I = sup ldv'f I·
Br/E:j
Here we abbreviate Bp = Bp(O).(b) The energy of vj outside the ball of radius 1 is bounded by li/2. More
precisely, for every R > 0 there exists a VR E N such that E( vj, BR - Bi) :::;
li/2 for v 2: VR. This implies that no bubbling can occur outside the unitball and hence the derivative of vj on the annulus BR - Bl+E is uniformly
bounded for every R > 0 and every c > 0.
(c) The sequence vj converges, modulo further bubbling, to a J-holomorphic
sphere v 1 : <CU { oo} __, M. Moreover, the image of the limit sphere v 1 isconnected to image of the original limit curve u. Namely, v 1 (oo) = u(z 1 ). A
detailed proof can be found in [19, 31, 37].