Conformally Invariant Variational Problems. - SAM

Theorem X.14. Let ⃗ Φ be a Lipschitz conformal immersion of
the disc D 2 with L 2 −bounded second fundamental form. Assume
⃗Φ is a weak Willmore immersion then there exists ⃗ L ∈ L 2,∞
loc (D2 )
such that
∇ ⊥ ⃗ L = ∇ ⃗ H −3π⃗n (∇ ⃗ H)+⋆(∇ ⊥ ⃗n∧ ⃗ H) ,
(X.186)
where ⃗n and ⃗ H denote respectively the Gauss map and the mean
curvature vector associed to the immersion ⃗ Φ.
Moreover the following conservation laws are satisfied
and
div < ⃗ L,∇ ⊥ ⃗ Φ >= 0 ,
[ ]
div ⃗L∧∇ ⊥⃗ Φ+2 (⋆(⃗n H)) ⃗ ∇
⊥⃗ Φ
(X.187)
= 0 . (X.188)
where ⋆ is the ususal Hodge operator on multivectors for the
canonical scalar product in R m and is the operations between
p− and q− vectors (p ≥ q) satisfying for any α ∈ ∧ p R m , β ∈
∧ q R m and γ ∈ ∧ p−q R m
< α β,γ >=< α,β ∧γ > .
Proof of theorem X.14. Let
( 1
[
] )
⃗F := div
2π log r ⋆χ ∇ ⊥ H ⃗ −3π⃗n (∇ ⊥ H)−⋆(∇⃗n∧ ⃗ H) ⃗
where χ is the characteristic function of the disc D 2 . Under the
assumptions of the theorem we have seen that
]
[∇ ⊥ H ⃗ −3π⃗n (∇ ⊥ H)−⋆(∇⃗n∧ ⃗ H) ⃗ ∈ L 1 loc ∩H−1 loc (D2 )
Hence a classical result on Riesz potentials applies (see [Ad]) in
order to deduce that
⃗F ∈ L 2,∞
loc (D2 ) .
185
✷