Conformally Invariant Variational Problems. - SAM

almost the information we started from 54 . This phenomenon
characterizes critical elliptic systems as we saw it already
in the first section of this course while presenting the elliptic
systems of quadratic growth in two dimension for W 1,2 norm.
It remains now in this subsection to prove theorem X.7.
In order to make the proof of theorem X.7 more accessible
we first give a proof of it in the codimension 1 setting which is
simpler and then we will explain how to generalize it to higher
codimension.
The result is a local one on Σ 2 therefore we can work locally
inadisc-neighborhoodofapointanduseisothermalcoordinates
onthisdisc. Thismeansthatwecanassume ⃗ Φtobeaconformal
immersion from the unit disc D 2 ⊂ R 2 into R 3 .
we will need the following general lemma for conformal immersions
of the 2-disc in R 3
Lemma X.2. Let Φ ⃗ be a conformal immersion from D 2 into R 3 .
Denote by ⃗n the Gauss map of the conformal immersion Φ ⃗ and
denote by H the mean curvature. Then the following identity
holds
−2H ∇Φ ⃗ = ∇⃗n+⃗n×∇ ⊥ ⃗n (X.91)
where ∇· := (∂ x1·,∂ x2·) and ∇ ⊥· := (−∂ x2·,∂ x1·).
Proof of lemma X.2. Denote (⃗e 1 ,⃗e 2 ) the orthonormalbasis
of ⃗ Φ ∗ (TΣ 2 ) given by
⃗e i := e −λ ∂⃗ Φ
∂x i
,
54 We will see in the next subsection that ⃗n ∈ W 1,2 implies that the conformal factor is
bounded in L ∞ and then, since again the parametrization is conformal, we have
.
H ∈ L 2,∞
loc (D2 ) =⇒ ∇⃗n ∈ L 2,∞
loc (D2 )
✷
146