LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 45such that a parametrizes a punctured neighbourhood of z 0. Define
ii: [O,oo) x 81 ---+ JR x M
ii= u o a.
Then(3)Vs+ Jiit = 0 on [O, oo) x 81E(ii) < oo.
An important result is now, writing S = 8 \ f:
Theorem 1.24. Assume ii solves (3). Then with ii= (b, v)
m(z 0 ) := lim { v(s,.)* ..\
S->00 J 51exists. We call m(zo) the mass at zo. If m(zo) = 0 then the map u: S---+ JR x M
can be smoothly extended over zo. If m(z 0 ) i- 0 then any given sequence sk---+ +oo
has a subsequence, also denoted by (sk) such that
lim v(sk, t) = x(m(zo)t)
k->oo
exists in C^00 (8^1 , M), where x solves
x = X(x).In particular x is lm(zo)l-periodic.
For a proof see [40].
This theorem has the following corollary:Corollary 1.25. Assume ..\, J, ~ and X are the data on M. If there exists a (non-
constant) finite energy surface u: S---+ JR x M, then X has a periodic solution.
Proof. We know that S = 8 \ r with "r 2: l. This is a consequence of the previous
exercise. If for every zo E r the mass m(zo) = 0 vanishes, all the punctures are
removable and u: 8 ---+ JR x M, with 8 closed. But then u = const which implies
E(u) = 0. Therefore there exists z 0 Er with m(z 0 ) i- 0 and the result follows from
Theorem 1.24. D
Theorem 1.21 is now a consequence of the following existence result for finite
energy planes, which are one-punctured finite energy spheres.
Theorem 1.26. Let ..\ be a contact form on the closed three-manifold M. Let J
be an admissible complex multiplication for ~ ---+ M and J the associated almost
complex structure on JR x M. If M = 83 or 7r 2 ( M) i- 0 or ~ is overtwisted there
exists a nontrivial finite energy plane u: C ---+ JR x Mus+ Jut= 0
0 < E(u) < oo.
As a corollary we obtain a criterion for the tightness of a contact structure.Corollary 1.27. If~ is a contact structure on a closed orientable three manifold
M induced by a contact form..\ admitting no contractible periodic orbit (x, T) then
~ is tight.