Conformally Invariant Variational Problems. - SAM

Let TR m ⃗ Φ([0,1]×Σ 2 ) be the restrictionof the tangentbundle
to R m over ⃗ Φ([0,1]×Σ 2 ).
The pull-back bundle ⃗ Φ −1 (TR m ⃗ Φ([0,1] × Σ 2 )) can be decomposed
into a direct bundle sum
⃗Φ −1 (TR m ⃗ Φ([0,1]×Σ 2 )) = T ⊕ N
↓
I × Σ
where the fiber T (t,p) over the point (t,p) ∈ [0,1]×Σ 2 is made
of the vectors in R m tangent to ⃗ Φ t (Σ 2 ) and N (t,p) is made of the
vectors in R m normal to ⃗ Φ ∗ (T (t,p) Σ 2 ) at ⃗ Φ(t,p). On T we define
the connection ∇ as follows : let σ be a section of T then we set
∀X ∈ T (t,p) ([0,1]×Σ 2 ) ∇ X σ := π T (dσ ·X) .
where π T is the orthogonal projection onto ⃗ Φ ∗ (T (t,p) Σ 2 ). On N
we define the connection D as follows : let τ be a section of N
then we set
∀X ∈ T (t,p) ([0,1]×Σ 2 ) D X σ := π ⃗n (dσ ·X) .
where π ⃗n is the orthogonal projection onto the normal space to
⃗Φ ∗ (T (t,p) Σ 2 ).
The first part of the proof consists in computing
D ∂
⃗H(0,p) ∀p ∈ Σ 2 .
∂t
To this purpose we introduce in a neighborhood of (0,p) some
special trivialization of T and N.
Let (⃗e 1 ,⃗e 2 ) be a positive orthonormal basis of ⃗ Φ ∗ (T (0,p) Σ 2 ).
Both ⃗e 1 and ⃗e 2 are points in the bundle T. We first transport
parallely ⃗e 1 and ⃗e 2 , with respect to the connection ∇, along the
path {(t,p) ; ∀t ∈ [0,1]}. The resulting parallely transported
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