CHAPTER 2
An outline of the paper
Due to the complexity of the paper, an overview of the ideas and the organi-
zation of the paper is necessary before an in-depth reading.
2.1. The frame method and the Coulomb gauge
At first sight the Schr¨ odinger Map equation has little to do with the Schr¨odinger
equation. A good way to bring in the Schr¨ odinger structure is by using the frame
method. Precisely, at each point (x, t)
R2+1
one introduces an orthonormal frame
(v, w) in
Tu(x,t)S2.
This frame is used to measure the derivatives of u, and reexpress
them as the complex valued radial differentiated fields
ψ1 = ∂ru · v + i∂ru · w, ψ2 = ∂θu · v + i∂θu · w.
Here the use of polar coordinates is motivated by the equivariance condition. Thus
instead of working with the equation for u, one writes the evolution equations for
the differentiated fields. The frame (v, w) does not appear directly there, but only
via the real valued radial connection coefficients
A1 = ∂rv · w, A2 = ∂θv · w, A0 = ∂tv · w.
A-priori the frame is not uniquely determined. To fix it one first asks that
the frame be equivariant, and then that it satisfies an appropriate condition. Here
it is convenient to use the Coulomb gauge; due to the equivariance this takes a
very simple form, A1 = 0. The construction of the Coulomb gauge is the first
goal in the next section. In Proposition 3.2 we prove that for
˙
H
1
equivariant maps
into
S2
close to Q there exists an unique Coulomb frame (v, w) which satisfies
appropriate boundary conditions at infinity, see (3.17). In addition, this frame has
a
C1
dependence on the map u.
In the Coulomb gauge the other spatial connection coefficient A2, while nonzero,
has a very simple form A2 = u3. We will also compute A0 in terms of ψ1, ψ2 and
A2,
(2.1) A0 =
1
2
|ψ1|2

1
r2
|ψ2|2
+ [r∂r
−1] |ψ1|2

1
r2
|ψ2|2
2.2. The reduced field ψ
Due to the equivariance the two fields ψ1 and ψ2 are not independent. Hence
it is convenient to work with a single field
ψ = ψ1
ir−1ψ2
7
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