1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. ANALYTICAL TOOLS 59

The third assertion gives a characterisation of completeness which is a version

of Ekeland's variational principle. It is very useful in the calculus of variations,

see [33]. The second assertion is due to Hofer and very useful in carrying out

an analysis for conformally invariant problems on unbounded domains. This will

become clear in the following bubbling-off analysis.

Exercise 2.7. Prove proposition 2.6.

Now we are in the position to prove the theorem. Take a sequence Ee ---+ 0 such

that

IY'ue(ze)IEe---+ + oo.

Applying the lemma and perhaps replacing ze by z~ and Ee by E~ we may assume

in addition to ( 4) that

(5)

IY'ue(z)I ::::; 2IY'ue(ze)I, lz - zel ::::; Ee

IY'ue(ze)IEe---+ +oo

Ee ---+ 0.

Define ue = (ae, ue) and

Then

z z

ve(z)=(ae(ze+ Re)-ae(ze),ue(ze+ Re))

Re = IY'ue(ze)I.

8/ve + l(ve)8tve = O

E( ve) ::::; c

IY've(z)I::::; 2

IY've(O)I = 1.

on DRe(l-lzel)

on DRe<e

Hence, by the C^00 -Ascoli-Arzela theorem, after taking a subsequence

ve---+ v in C 1 ~(C, JR x M)

osv + lotv = 0 on C

E(v) ::::; c

IY'v(O) I = 1.

Therefore v : C ---+ JR x M is non-constant and

Vs+ lvt = 0

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

A much more sophisticated bubbling-off analysis is possible and we refer the
reader to the literature, [48, 49, 51, 55]. For the related bubbling-off analysis in
the compact case see [36, 42, 75, 88, 90].

2.3. Behaviour near a puncture

Using holomorphic polar coordinates near the puncture we may assume

u: [O,oo) x S^1 ---+ JR x M

(6) Us+ Jut= 0

E(u) < oo.