Conformally Invariant Variational Problems. - SAM

where ⃗ I and ⃗n denote respectively the second fundamental form
andtheGaussmapoftheconformalimmersionΦ◦Ψ. ⃗ Themean
curvature vector of Φ◦Ψ, ⃗ that we simply denote H, ⃗ is given by
⃗H [ := 2 −1 e −2λ ⃗ I(∂x1 ,∂ x1 )+ ⃗ ]
I(∂ x2 ,∂ x2 ) ∈ L 2 (D 2 ) .
Hence we have that
moreover, we have also
∇ ⃗ H ∈ H −1 (D 2 ) ,
⋆(∇ ⊥ ⃗n∧ ⃗ H) ∈ L 1 (D 2 ) .
Using the expression (X.88), we also deduce that
π ⃗n (∇ ⃗ H) ∈ H −1 +L 1 (D 2 ) .
Hence for any immersion ⃗ Φ in E Σ the quantity
∇ ⃗ H −3π ⃗n (∇ ⃗ H)+⋆(∇ ⊥ ⃗n∧ ⃗ H) ∈ H −1 +L 1 (D 2 )
defines a distribution in D ′ (D 2 ).
Using corollary X.4 one can then generalize the notion of
Willmore immersion that we defined for smooth immersions to
immersions in E Σ in the following way.
Definition X.4. Let Σbe asmoothcompactoriented2-dimensional
manifold and let Φ ⃗ be a Lipschitz immersion with L 2 −bounded
second fundamental form.
⃗Φ is called a weak Willmore immersion if, in any lipschitz
conformal chart Ψ from D 2 into (Σ, Φ ⃗ ∗ g R m), the following holds
[
div ∇H ⃗ −3π ⃗n (∇H)+⋆(∇ ⃗ ⊥ ⃗n∧ H) ⃗ ]
= 0 in D ′ (D 2 ) .
where ⃗ H and ⃗n denote respectively the mean curvature vector
and the Gauss map of ⃗ Φ in the chart Ψ and the operators div,
∇ and ∇ ⊥ are taken with respect to the flat metric 65 in D 2 . ✷
65 divX = ∂ x1 X 1 +∂ x2 X 2 , ∇f = (∂ x1 f,∂ x2 f) and ∇ ⊥ f = (−∂ x2 f,∂ x1 f).
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