Conformally Invariant Variational Problems. - SAM

Introduce (φ 1 ,φ 2 ) the functions with average 0 on the disc D 2
such that
dφ i := e −λ e ∗ i .
since rank(dφ 1 ,dφ 2 ) = 2, φ := (φ 1 ,φ 2 ) realizes a diffeomorphism
from D 2 into φ(D 2 ).
From the previous identity we have
e −λ◦φ−1 g(e j ,∂ yi φ −1 ) = e −λ◦φ−1 (e ∗ j ,∂ y i
φ −1 ) = δ ij .
where g := ⃗ Φ ∗ g R m. This implies
or in other words
g(∂ yi φ −1 ,∂ yj φ −1 ) = e 2λ◦φ−1 δ ij . (X.142)
< ∂ yi ( ⃗ Φ◦φ −1 ),∂ yj ( ⃗ Φ◦φ −1 ) >= e 2λ◦φ−1 δ ij . (X.143)
This says that ⃗ Φ ◦ φ −1 is a conformal immersion from φ(D 2 )
into R m . The Riemann Mapping theorem 58 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 ).
X.6.2 The space of Lipschitz Immersions with L 2 −bounded Second
Fundamental Form.
In the previoussubsection we haveseen the equivalence between
tangent Coulomb moving frames and isothermal coordinates. It
remains now to construct tangent Coulomb moving frames in
order to produce isothermal coordinates in which Willmore surfaces
equation admits a nice conservative elliptic form. For a
purpose that will become clearer later in this book we are extending
the framework of smooth immersions to a more general
framework: thespaceofLipschitz immersions with L 2 −bounded
second fundamental form.
58 See for instance [Rud] chapter 14.
162