Conformally Invariant Variational Problems. - SAM

Let Σ 2 be a smoothcompactoriented2-dimensionalmanifold
(withorwithoutboundary). Letg 0 beareferencesmoothmetric
onΣ. OnedefinestheSobolevspacesW k,p (Σ,R m )ofmeasurable
maps from Σ into R m in the following way
{
k∑
∫ }
W k,p (Σ 2 ,R m ) = f : Σ 2 → R m ; |∇ l f| p g 0
dvol g0 < +∞
Since Σ 2 is assumed to be compact it is not difficult to see that
this space is independent of the choice we have made of g 0 .
AlipschitzimmersionofΣ 2 intoR m isamap ⃗ ΦinW 1,∞ (Σ 2 ,R m )
for which
l=0
Σ
∃ c 0 > 0 s.t. |d ⃗ Φ∧d ⃗ Φ| g0 ≥ c 0 > 0 ,
(X.144)
whered ⃗ Φ∧d ⃗ Φisa2-formonΣ 2 takingvaluesinto2-vectorsfrom
R m and given in local coordinates by 2∂ x1
⃗ Φ ∧ ∂x2
⃗ Φ dx1 ∧ dx 2 .
The condition (V.4) is again independent of the choice of the
metric g 0 . This assumption implies that g := ⃗ Φ ∗ g R
m defines an
L ∞ metric comparable to the reference metric g 0 : there exists
C > 0 such that
∀X ∈ TΣ 2
C −1 g 0 (X,X) ≤ ⃗ Φ ∗ g R m(X,X) ≤ C g 0 (X,X) .
(X.145)
For a Lipschitz immersion satisfying (X.144) we can define the
Gauss map as being the following measurable map in L ∞ (Σ)
⃗n ⃗Φ := ⋆ ∂ x 1
⃗ Φ∧∂x2 ⃗ Φ
|∂ x1
⃗ Φ∧∂x2 ⃗ Φ|
.
for an arbitrary choice of local positive coordinates (x 1 ,x 2 ). We
then introduce the space E Σ of Lipschitz immersions of Σ with
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