Conformally Invariant Variational Problems. - SAM

Since F ⃗ has been chosen in order to have
∆F ⃗ ]
= div
[∇ ⊥ H ⃗ −3π⃗n (∇ ⊥ H)−⋆(∇⃗n∧ ⃗ H) ⃗
, (X.189)
We introduce the distribution
⃗X := ∇ ⊥ F +∇ ⃗ H −3π ⃗n (∇ ⃗ H)+⋆(∇ ⊥ ⃗n∧ ⃗ H) .
Combining (X.78) and the fact that Φ ⃗ is Willmore, which is
equivalent to (X.86), we obtain that X ⃗ satisfies
⎧
⎨ divX ⃗ = 0
⎩
curl ⃗ X = 0
Hence the components of ⃗ X = (X 1 ,··· ,X m ) realize harmonic
vectorfields and there exists then an R m −valued harmonic map
⃗G = (G 1 ,··· ,G m ) such that
⃗X = ∇ ⊥ ⃗ G .
Being harmonic, the map ⃗ G is analytic in the interior of D 2 ,
therefore 72 ⃗ L := ⃗ G− ⃗ F is in L
2,∞
loc (D2 ) and satisfies (X.186).
We now establishthe first conservationlaw(X.187). We have
< ∇ ⃗ Φ,∇ ⊥ ⃗ L >=< ∇ ⃗ Φ,∇ ⃗ H > + < ∇ ⃗ Φ,⋆(∇ ⊥ ⃗n∧ ⃗ H) > .
Multiplying (??) by H α , summing over α = 1···m − 2 and
projecting over ⃗ Φ ∗ TD 2 using the tangential projection π T gives
π T (∇ ⃗ H −⋆(∇ ⊥ ⃗n∧ ⃗ H)) = −2 | ⃗ H| 2 ∇ ⃗ Φ .
(X.190)
Hence we have
< ∇ ⃗ Φ,⋆(∇ ⊥ ⃗n∧ ⃗ H) >=< ∇ ⃗ Φ,∇ ⃗ H > +2 | ⃗ H| 2 |∇ ⃗ Φ| 2 .
72 In fact (X.186) is telling us that ∇ ⃗ L belongs to H −1 + L 1 (D 2 ) and using a more
sophisticated result (see [BoBr] theorem 4) one can infer that in fact ⃗ L ∈ L 2 (D 2 ).
186