Conformally Invariant Variational Problems. - SAM

isabilipschitzdiffeomorphismbetweenD 2 andφ(D 2 ). Equation
(X.143) says that ⃗ Φ◦φ −1 is a conformal lipshitz immersion from
φ(D 2 ) into R m . The Riemann Mapping theorem gives the existence
of a biholomorphic diffeomorphism h from D 2 into φ(D 2 ).
Thus ⃗ Φ◦φ −1 ◦h realizes a conformal immersion from D 2 onto
⃗Φ(D 2 ) which is in W 1,∞
loc (D2 ,R m ). We have then established the
following theorem.
Theorem X.12. [Existence of a smooth conformal structure]
Let Σ 2 be a closed smooth 2-dimensional manifold. Let ⃗ Φ
be an element of E Σ : a Lipshitz immersion with L 2 −bounded
second fundamental form. Then there exists a finite covering of
Σ 2 by discs (U i ) i∈I and Lipschitz diffeomorphisms ψ i from D 2
into U i such that ⃗ Φ ◦ ψ i realizes a lipschitz conformal immersion
of D 2 . Since ψ −1
j ◦ ψ i on ψ −1
i (U i ∩ U j ) is conformal and
positive (i.e. holomorphic) the system of charts (U i ,ψ i ) defines
a smooth conformal structure c on Σ 2 and in particular there
exists a constant scalar curvature metric g c on Σ 2 and a lipshitz
diffeomorphism ψ of Σ 2 such that Φ ⃗ ◦ ψ realizes a conformal
immersion of the riemann surface (Σ 2 ,g c ). ✷
X.6.5 Weak Willmore immersions
Let Σ be a smooth compact oriented 2-dimensional manifold
and let Φ ⃗ be a Lipschitz immersion with L 2 −bounded second
fundamental form : Φ ⃗ is an element of E Σ . Because of the previous
subsection we know that for any smooth disc U included in
Σ there exists a Lipschitz diffeomorphism from D 2 into U such
that Φ ⃗ ◦ Ψ is a conformal Lipschitz immersion of D 2 . In this
chart the L 2 − norm of the second fundamental form which is
assumed to be finite is given by
∫ ∫
| ⃗ I| 2 g dvol g = |∇⃗n| 2 dx 1 dx 2 .
U D 2
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