192 D. SALAMON, FLOER HOMOLOGY
( d) For every j,
(38) m(zj) = R~lim lim oo v-.oo E(vj, BR)·
(39)
(e) If the limit curve Vj is constant then, for every r > 1,
lim E(vj,Br-B1) = -
2!i.
V--+00This means that the energy !i/2 of the approximating curves vj in the domain
B r - B 1 is concentrated in an arbitrarily small neighbourhood of the unit
circle.Figure 12. The bubbling phenomenonTo prove (d) one can choose a sequence pj -7 0 such that E(u", Bpj(zj)) -7
m( Zj) and examine the quotient pj / cj. If ( 38) does not hold then this quotient
tends to infinity and one can prove that the energy of u" in the annulus Bpj (zj) -
Bc:j (zj), which converges to !i/2, is concentrated in the subset BRc:j (zj) -Bc:j (zj).
More precisely, there exists a constant c > 0 such that
lim E(u",B&dz 1 ") - Bc:~(zj)) 2: (i -
1cR) ~
2.
V--+ 00 J J ogBy (37), this implies
lim E(vj,BR) = lim E(u", BRc:dzj)) 2: m(zj)-
1
cR_
2
!i ,
l/--+00 V--+ 00 J Ogin contradiction to the assumption that ( 38) does not hold. This proves ( d). Full
details are given in [19, 31]. If v is constant then, by (b), the limit on the right
hand side of (38) is independent of R > l. Hence (39) follows from (d). In turn
it follows from (39) that, for the sequence vj, bubbling occurs on the unit circle.
By (a), this means that there is also a bubble at the origin. In summary, ifthe limit
curve Vj is constant then the sequence vj exhibits bubbling at two or more points
inside the unit ball. This has two crucial consequences. Firstly, since the total
mass of all the bubbles of vj is equal to the mass m( Zj) of the original sequence
u", it follows that, if this mass is divided among two bubbles, each part is at most
m(zj) - Ii. This enables us to carry out an induction argument, replacing u" by
vj, which must terminate after finitely many steps. Secondly, we observe that, if
the limit curve Vj is constant, it is connected at three or more points to other J-
holomorphic curves in the bubble tree. Namely, at z = oo it is connected to u, and
at z = 0 and some point on the unit circle it is connected to the next bubbles in
the induction argument. This observation leads naturally to the notion of a stable
map as a tree of J-holomorphic spheres, where each constant sphere is connected