208 D. SALAMON, FLOER HOMOLOGYThe solutions of this differential inequality will be called (J, r)-holomorphic
curves. To ensure that the space of such curves is still invariant under G we
shall assume that r is equivariant.
Suppose, for example, that an arbitrarily small multi-valued perturbation of
this form is introduced in a neighbourhood of a perfectly regular J-holomorphic
curve. Then this curve will split up into finitely many solutions of ( 4 7), depending
on the number of branches of r , and the union of these curves should be counted
as one curve. Alternatively, each of the individual solutions should be counted withsome weight so that the sum of the weights is l. This motivates the introduction
of positive rational weights A(v, ry) > 0 for ry E r(v) such that
.L A(v,ry) = 1
ryEI"(v)for every v EB. Call the pair (r, A) a weighted multi-valued G-perturbation.Such perturbations form a semigroup with sum (r, A) = (r 1 + r 2 , A 1 * A 2 ) defined
byr(v) = {rJ1 + 'r/ 2 : 'r/i E ri(v)}, A(v,ry) = L A1(v,ry1) · A2(v,ry2).
T/1 +T/2=1)
Here we use the convention A(v, ry) 0 for ry i r(v). With this convention, ris uniquely determined by A. In fact, one can think of (r, A) as a collection of
discrete measures on the fibers Ev of our infinite dimensional vector bundle. We
shall impose the following conditions on the multi-valued perturbations. The first
two were already mentioned above.
(Finiteness) r(v) is a finite subset of Ev for every v EB. The weight functionr(v) -t Q: ry f-+ A(v, ry) is positive and satisfies I::ryEI"(v) A(v, ry) = 1 for every
v EB.(Conformality) For v EB, ry E Ev, and <p E G
r(v 0 <p) = <p*r(v),
(Energy) There is a constant c = c(r) > 0 such that, for all v E B and all
rJ E Ev,rJ E r(v)(Local structure) For every u E B there exists a C^0 -neighbourhood U of u,
finitely many continuous sections Ii : U -t E, i = 1, ... , m , and positive
rational numbers A1, ... , Am such that A 1 + · · · +Am = 1 and
r(v) = b1(v), ... 'lm(v)}' A(v,ry) = L Ai
"'f;(v)=ry
for v E U and ry E Ev. The I i are called the branches of r and Ai is called
the weight of I i · We assume that the branches satisfy the following.- Each Ii restricts to a Ce-function from Uk,p -t Ek-t ,p for 0 ::; e ::; k.^1
(^1) The superscripts denote Sobo lev completions. Thus £~,p = W"',P(S (^2) , A 0 , (^1) T S (^2 181) vT M), and
U k,p =Un Bk,p is an open subset of Bk,p = W"''P(S^2 , M).