LECTURE 5. MULTI-VALUED PERTURBATIONS 223which is equivariant with respect to the JR-action on Z and E. Instead of (7) we
shall then consider the differential inequality
(55) aJ,H(u) = OsU + J(u)OtU - V' Ht(u) E r(u).
The axioms for rare as before "finiteness", "conformality" (here equivariance under
the JR-action), "local structure", and the energy bound
(56) T/ E f(u)
J
oo f 1 ITJl^2 dtds S c.
-oo lo
The "transversality" and "free" axioms are immaterial in this case. Transversality
can be achieved by a generic perturbation of H (see [13] or [47]), and that the
JR-action is free is obvious unless x-= x+, and this case can easily be dealt with
separately. Instead, the crucial point is the "compatibility" axiom, which is needed
to prevent the bubbling off of multiply covered J-holomorphic spheres with negative
Chern number as in Section 5.1. The purpose of the perturbation is indeed, to make
shure that what bubbles off are not J-holomorphic curves, but (J, r)-holomorphic
curves, because these form moduli spaces of the predicted dimensions. To carry
this out one must introduce a notion of stable connecting orbits in analogy to
the notion of stable maps, by including finite chains of connecting orbits together
with bubble trees. The stability condition allows the case of connecting orbits of
the form u(s, t) = x(t) for some x E P(H), but such a onnecting orbit must contain
at least one double point at which it intersects a J-holomorphic curve in the tree
(see Figure 22).
Figure 22. Stable connecting orbitsThen there is the notion of Floer-Gromov convergence for stable connecting
orbits, and the compatibility condition has the form of continuity with respect to
this Floer-Gromov topology. In other words, the perturbation for the stable
connecting orbits will involve perturbations for the J-holomorphic curves in the
bubble tree and if uv is a sequence of connecting orbits converging to such a stable
connecting orbit in the Floer-Gromov topology, then the perturbations f( uv) must
converge to those for the limit configuration. The details are as in Sections 4.2
and 5.2 and will not be discussed here. The reader is instead referred to [23].It is important to note that the differential inequality (55) is no longer a gradient
flow equation. Moreover, the energy identity of Exercise 1.23 now becomes an
energy inequality.