1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

58 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS

2.2. Bubbling-off analysis

Assume we are given a sequence iie: D ___, IR x M of smooth maps satisfying

(u e)s + ](ue)t = O

(4)

E(ue) :::; c

l\lue(ze)I ___, oo

ze ___, 0.

Recall that the energy E(u) is defined by

E(u) = supcpEE k u*d(cp>..),

where ~ consists of all smooth maps cp : IR ___, [O, l] satisfying cp' ( s) 2 0 for all s E R

Theorem 2.4. Let a sequence be given as in ( 4). Then there exists a finite energy
plane

We need the following

u:C---+IRxM

Us + Jut= 0

0 < E(u):::; c.

Lemma 2.5. Let (Y, d) be a complete metric space and cp: Y ___, JR+ a continuous

map. Let E > 0 be given and x 0 E Y. Then there exists E^1 E (0, E] and x~ with

d(xo, x~) :::; 2E

cp(x~)t:' 2 cp(xo)E

cp(x):::; 2cp(x~) for d(x, x~) :::; t:'.

This lemma may be viewed as a corollary of the following topological proposi-

tion which is quite useful.

Proposition 2.6. Let (Y, d) be a metric space. The following assertions are equiv-
alent.

1. The metric space (Y, d) is complete.

2. For every continuous map cp : Y ___, [O, oo), given E > 0 and x 0 E Y there

exists E^1 E (0, E] and x~ with

d(xo, x~) :::; 2E

cp(x~)E^1 2 cp(xo)E

cp(x):::; 2cp(x~) for d(x,x~):::; E^1 •

3. Given any continuous map cp : Y ___, [O, oo) there exists for given c > 0,

A > 0 and a point x 0 satisfying

a point y such that

cp(xo):::; c + infxEY cp(x)

cp(y) :::; cp(x)

d(x, y):::;±

cp(z ) 2 cp(y) - E:Ad(y, z) for all z E Y.