Conformally Invariant Variational Problems. - SAM

such a way that ⃗n ρ λ realizes a degree one map from S2 into S 2 .
Any degree one map from S 2 into S 2 is homotopic to the identity
map and the pull-back bundle ((⃗n ρ λ )−1 ) −1 TS 2 is then bundle
equivalent to TS 2 (see theorem 4.7 in chapter 1 of [Hus]).⃗e i
wouldrealizeaglobalsectionofthisbundlethatwouldbetrivial
which would contradicts Brouwer’s theorem. Hence the restriction
to ∂D 2 of ⃗e i has a non zero degree 63 and by homotopy this
is also the case on any circle ∂B r (0) for 1 > r > ρ. Since ⃗e 1 has
non zero degree on each of these circles one has
∫
[∫ ] 1/2
∀ρ < r < 1 2π ≤
∣ (⃗e 1 ) ∗ dθ
∣ ≤ (2πr)1/2 |∇⃗e 1 | 2
∂B 2 r
We deduce from this inequality for i = 1,2
∫
|∇⃗e i | 2 dx 1 dx 2 ≥ 2πlog 1 → +∞ as ρ → 0 . (X.152)
D ρ 2
By taking λ → +∞ and ρ → +∞, we can deduce the following
lemma.
Lemma X.6. Thereexistsasequence⃗n k inW 1,2 (D 2 ,Gr m−2 (R m ))
such that ∫
|∇⃗n k | 2 dx 1 dx 2 −→ 8π
D 2
and
⎧ ∫
⎪⎨
2∑
|∇⃗e i | 2 dx 1 dx 2 s.t.
inf D
⎪⎩
2
i=1
⎪⎭ → +∞ ⃗e i ∈ W 1,2 (D 2 ,S m−1 ) and ⃗e 1 ∧⃗e 2 = ⋆⃗n
✷
⎫
⎪⎬
∂B 2 r
Thislemmasaysthatit isnecessary tostaystrictlybelowthe
threshold 8π for the Dirichletenergy of mapsinto Gr m−2 (R m ) in
63 This degree is in fact equal to 2 which is the Euler characteristic of S 2 . See theorem
11.16 of [BoTu] and example 11.18.
167