Conformally Invariant Variational Problems. - SAM

and consequently
⋆(⃗n∧∇ ⊥ ⃗n α ) =< ∇ ⊥ ⃗n α ,⃗e 1 > ⃗e 2 − < ∇ ⊥ ⃗n α ,⃗e 2 > ⃗e 1 (X.107)
Hence
⋆(∇ ⊥ ⃗n∧ H) ⃗ = − < ∇ ⊥ H,⃗e1 ⃗ > ⃗e 2 + < ∇ ⊥ H,⃗e2 ⃗ > ⃗e 1
=< ⃗ H,π ⃗n (∇ ⊥ ⃗e 1 ) > ⃗e 2 − < ⃗ H,π ⃗n (∇ ⊥ ⃗e 2 ) > ⃗e 1
Using (X.104), we then have proved
⋆(∇ ⊥ ⃗n∧ H) ⃗ =
⎛
⎝ − < H, ⃗ ⃗ ⎞
h 12 > ∂ x2Φ+ ⃗ < H, ⃗ ⃗ h22 > ∂ x1Φ
⃗
⎠
< H, ⃗ ⃗ h 11 > ∂ x2Φ− ⃗ < H, ⃗ ⃗ h12 > ∂ x1Φ ⃗
(X.108)
The tangential projection of ∇ ⃗ H is given by
π T (∇ ⃗ H) =< ∇ ⃗ H,⃗e 1 > ⃗e 1 + < ∇ ⃗ H,⃗e 2 > ⃗e 2
= − < ⃗ H,π ⃗n (∇⃗e 1 ) > ⃗e 1 − < ⃗ H,π ⃗n (∇⃗e 2 ) > ⃗e 2 .
Hence
π T (∇H) ⃗ =
⎛
⎝ − < H, ⃗ ⃗ ⎞
h 11 > ∂ x1Φ− ⃗ < H, ⃗ ⃗ h12 > ∂ x2Φ
⃗
⎠
− < H, ⃗ ⃗ h 12 > ∂ x1Φ− ⃗ < H, ⃗ ⃗ h22 > ∂ x2Φ ⃗
(X.109)
Combining (X.108) and (X.109) gives
−π T (∇ ⃗ H)−⋆(∇ ⊥ ⃗n∧ ⃗ H) =
⎛
⎝ < H, ⃗ ⃗ h 11 − ⃗ ⎞
h 22 > ∂ x1Φ+2 ⃗ < H, ⃗ ⃗ h12 > ∂ x2Φ
⃗
⎠ (X.110)
2 < H, ⃗ ⃗ h 12 > ∂ x1Φ+ ⃗ < H, ⃗ ⃗ h22 − ⃗ h 11 > ∂ x2Φ ⃗
151