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

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