1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. FLOER HOMOLOGY 175

E(v) ~ n. Now the subsequences (still denoted by u" and v") satisfy the following,

for every (small) c: > 0 and every (large) R > 0,

lim inf E( u", B 0 (z)) > lim inf E(u", BR 0 v (z"))

V--+OO l/--+00

lim V-->CX) inf E( v", BR(O))

E(v, BR(O)).

Taking the limit R ~ oo we find

liminfE(uV-->CX) ",B 0 (z)) ~ n

for every c: > 0. Since the energy of u" is uniformly bounded above, this can only

happen at finitely many points z 1 , ... , ze. We can then choose a further subsequence

which converges, uniformly with all derivatives on compact subsets of IR x 81 -

{z1, ... , z£}, to a function u : IR x 81 - { z 1 , ... , ze} ~ M. The limit necessarily

satisfies (7) and

E ( u,IR x 8

1

-yB 0 (zi)) }~~ E ( u",IR x 81 - yBc(zi))

< limsupE(u")-LliminfE(u",B 0 (zi))

V--+00 t. V--+00

< limsupE(u")-en

V-->CX)

for every c: > 0. Taking the limit c: ~ 0 we find

E(u) ::::; limsup E(u") - en.

V-->CX)

Finally, the removable singularity theorem for solutions of (7) shows that u extends
to a smooth function on all of IR x 81. This proves the proposition. 0

The convergence in the complement of a finite set as in the assertion of Propo-
sition 3.3 will be called convergence modulo bubbling. The limit solution u
has finite energy and hence, by Proposition 1.21, is again a connecting orbit. One
can now argue as in the finite dimensional case (Exercise 1.9) to prove the following
corollary (see Figure 7).

Corollary 3.4. Suppose that the periodic solutions x E P(H) are all nondegenerate

and let u" E M(x-, x+; H, J) be a sequence which satisfies (25). Then there exist a

subsequence (still denoted by u" ), finitely many periodic orbits x 0 ,... , Xm E P(H)

with x 0 = x+ and Xm = x-, finitely many connecting orbits

j = l, ... , m,

and finitely many sequences sj, such that u" ( s + sj, t) converges modulo bubbling

to UJ(s, t). Moreover, the limit solutions satisfy

m
(27) LE(uJ)::::; limsupE(u")-en.
j=O V-->CX)

where e is the total number of bubbles.