Conformally Invariant Variational Problems. - SAM

We have seen how any conformal parametrization generate a
Coulomb frame in the tangent bundle. S.S.Chern observed that
this is in fact an exact matching, the reciproque is also true
: starting from a Coulomb frame one can generate isothermal
coordinates.
Let ⃗ Φ be an immersion of the disc D 2 into R m and let (⃗e 1 ,⃗e 2 )
be a Coulomb tangent orthonormal moving frame : a map from
D 2 into ⃗ Φ ∗ (TD 2 ×TD 2 ) such that (⃗e 1 ,⃗e 2 )(x 1 ,x 2 ) realizes a positive
orthonormal basis of ⃗ Φ ∗ (T (x1 ,x 2 )D 2 ) and such that condition
(X.129) is satisfied.
Let λ be the solution of
⎧
⎪⎨
⎪⎩
dλ = ∗ g
∫
∂D 2 λ = 0
(X.133)
Denote moreover e i := d ⃗ Φ −1·⃗e i and (e ∗ 1,e ∗ 2) to be the dual basis
to (e 1 ,e 2 ). The Cartan formula 56 for the exterior differential of
a 1-form implies
de ∗ i(e 1 ,e 2 ) = d(e ∗ i(e 2 ))·e 1 −d(e ∗ i(e 1 ))·e 2 −e ∗ i([e 1 ,e 2 ])
= −e ∗ i ([e 1,e 2 ])
= −(( ⃗ Φ −1 ) ∗ e ∗ i )([d⃗ Φ·e 1 ,d ⃗ Φ·e 2 ])
= −(( Φ ⃗ −1 ) ∗ e i )([⃗e 1 ,⃗e 2 ])
(X.135)
The Levi-Civitaconnection∇onΦ ⃗ ∗ TD 2 issued fromtherestriction
to the tangent space to Φ(D ⃗ 2 ) of the canonical metric in
56 The Cartan formula for the exterior differential of a 1 form α on a differentiable
manifolfd M m says that for any pair of vector fields X,Y on this manifold the following
identity holds
dα(X,Y) = d(α(Y))·X −d(α(X))·Y −α([X,Y]) (X.134)
see corollary 1.122 chapter I of [GHL].
160