Conformally Invariant Variational Problems. - SAM

X.7.4 The conformal Willmore surface equation.
As we have seen conformal immersions which are solutions to
the conservation laws (X.229) for some ⃗ L in L 2,∞ (D 2 ) satisfy an
elliptic system, the system (X.223) which formally resembles to
the CMC equation from which we deduced the smoothness of
the immersion. Then comes naturally the question
Are solutions to the conservation laws (X.229) Willmore ?
The answer to that question is ”almost”. We shall see below
that solutions to (X.229) for some ⃗ L in L 2,∞ (D 2 ) satisfy the
Willmore equation up to the addition of an holomorphic function
times the Weingarten operator. Assuming that for some
immersion ⃗ Φ of a given abstract surface Σ 2 (X.229) holds in any
conformal chart, then the union of these holomorphic functions
can be ”glued together” in order to produce an holomorphic
quadratic differential of the riemann surface Σ 2 equipped with
theconformalclassgivenby ⃗ Φ ∗ g R m. Aswewillexplainbelow,the
intrinsic equation obtained corresponds to the equation satisfied
by critical points to the Willmore functional but with the constraint
that the immersion realizes a fixed conformal class. As
we will see this holomorphic quadratic differential plays the role
of a Lagrange multiplier.
First we prove the following result.
Theorem X.18. Let Φ ⃗ be a conformal lipschitz immersion of
the disc D 2 with L 2 −bounded second fundamental form. There
exists L ⃗ in L 2,∞ (D 2 ,R m ) satisfying
⎧
⎪⎨ div < L,∇ ⃗ ⊥ Φ ⃗ >= 0
[ ] (X.239)
⎪⎩ div ⃗L∧∇ ⊥⃗ Φ+2 (⋆(⃗n H)) ⃗ ∇
⊥⃗ Φ = 0 .
if and only if there exists an holomorphic function f(z) such
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