Conformally Invariant Variational Problems. - SAM

where (g ik ) is the inverse matrix to (g ij ). Thus
√
d
dt
det(gij t )(0) 2∑
√ = g jk < ∂ xkV,∂xj ⃗ Φ ⃗ > .
det(gij )
k=1
Since we are led to consider only V ⃗ orthogonal to ( Φ ⃗ 0 ) ∗ (TΣ 2 )
we have
2∑
2∑
g jk < ∂ xkV,∂xj ⃗ Φ ⃗ >= − g jk < V,∂ ⃗ x 2 ⃗
j x kΦ >
k=1
= −
k=1
2∑
g jk < V,π ⃗ ⃗n (∂x 2 ⃗
j x kΦ) >= −2 < H, ⃗ V ⃗ > .
k=1
Hence we have proved that
d
dt (dvol ⃗ Φ
∗
t g R m )(0) = −2 < ⃗ H, ⃗ V > dvol ⃗Φ∗ g R m
. (X.84)
Combining (X.83) and (X.84) we obtain 52
∫
d
| H
dt
∫Σ ⃗ t | 2 dvol ⃗Φ ∗ 2 t g = 2 R
< D
m ∂tH, ⃗ H ⃗ > dvol⃗Φ∗
g R
Σ 2 m
∫
−2 | H| ⃗ 2 < H, ⃗ V ⃗ > dvol ⃗Φ∗ g R
Σ 2 m
=
〈∆ ⊥V ⃗ + Ã( V)−2|
∫Σ ⃗ ⃗ 〉
H| 2 V, ⃗ H ⃗ dvol ⃗Φ∗ g R 2 m
=
〈∆ ⊥H ⃗ + Ã( H)−2|
∫Σ ⃗ ⃗ 〉
H| 2 H, ⃗ V ⃗ 2
The immersion Φ ⃗ is Willmore if and only if
∫
d
| H
dt
⃗ t | 2 dvol ⃗Φ ∗
Σ 2 t g = 0
R m
dvol ⃗Φ∗ g R m
52 These first variations of ⃗ H and dvolg were known in codimension 1 probably even
before W.Blaschke see 117 in [Bla1].
138