1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

0 < E(u) < oo,

where E(u) is defined by

E(u) = sup<pEE r u*d(

.o).

ls\r

Here I; consists of all smooth maps <p : IR---+ [O, 1] satisfying <p^1 :::: 0 and <pAo

is the 1-form on IR x S^3 defined by

(cp>.o)(a, u)(h, k) = cp(a)>.o(u)(k).

We call E( u) the energy. So, identifying C^2 \ { 0} with IR x S^3 , we can charac-

terize affine algebraic sets as the images of solutions of a certain Cauchy Riemann

type problems, which satisfy some energy condition.

This leads immediately to some striking generalisations. Given a closed three-

manifold M and a contact form >. denote by ~ and X the associated contact struc-

ture and Reeb vector field, respectively. Pick a complex multiplication J : ~ ---+ ~

compatible with d>., i.e. d>.(h, Jh) > 0 for h E ~ \ {O}. Then define an almost

complex structure J on IR x M by

i(a, u)(h, k) = (->.(u)(k), J(u)7rk + hX(u)),

where 7r: TM---+~ is the projection along X.

Then we generalize our discussion of affine algebraic sets as follows. We are

looking for solutions of the differential equation

u:S\I'--+IRxM

Tuoj =lo Tu

0 < E(u) < oo.

Here the energy is defined as before by

E(u) = sup<pEE r u*d(cp>.)

ls\r

and (S,j) is a closed Riemannian surface with r being a finite subset of S. The

above system is elliptic. A analytical complication comes from the fact that the
domains are open. The energy condition implies that the behaviour near the punc-
tures is special. We shall see this in more detail.

Exercise 1.23. If ($, j) is a closed Riemannian surface and u : S ---+ IR x M
satisfying

Tuoj = J oTu,

then u is constant.

In order to understand the behaviour of u near a puncture let z 0 E r and pick
a holomorphic map

(]": [O,oo) x S^1 ---+ S