Conformally Invariant Variational Problems. - SAM

Assume that at p the vector H ⃗ ⃗Φ is parallel to ⃗n 1 , one has, using
the coordinates s we introduced,
∫
∣
∣< G, ⃗ H ⃗ ∫
⃗Φ > ∣ 2 Ω = s 2 1 |⃗ H ⃗Φ | 2 ω S
m−3
S m−3
Fiber
= |⃗ H ⃗Φ | 2
m−2
m−2
∑
j=1
∫S m−3 s 2 j ω S
m−3 = |⃗ H ⃗Φ | 2
m−2
Denote N + Σ 2 the subset of NΣ 2 on which det
non-negative. We have then proved
∫
G ⃗ ∗ ω S
m−1 ≤ 1 ∫
N + Σ 2π
2
.
(
< G, ⃗ ⃗ )
I >
Σ 2 | ⃗ H ⃗Φ | 2 dvol ⃗Φ∗ g R m
. (X.57)
Now we claim that each points in S m−1 admits at least two
preimages by G ⃗ (
at which det < G, ⃗ ⃗ )
I > ≥ 0 unless the immersed
surface is included in an hyperplane of R m .
Indeed, let ξ ⃗ ∈ S m−1 and consider the affine hyper-plane Ξ a
given by < x, ξ ⃗ >= a Let
and
a + = max{a ∈ R ; Ξ a ∩ ⃗ Φ(Σ 2 ) ≠ ∅}
a − = min{a ∈ R ; Ξ a ∩ ⃗ Φ(Σ 2 ) ≠ ∅}
Assume a + = a − then the surfaced is immersed in the hyperplane
Ξ a+ = Ξ a− and thus the claim is proved. If a + > a − , it is
clear that ⃗ Φ(Σ 2 ) is tangent to Ξ a+ at a point ⃗ Φ(p + ) and hence
⃗ξ belongs to the normal space to ⃗ Φ ∗ T p Σ 2 which means in other
words that ξ ⃗ ∈ G(N ⃗ p+ Σ 2 ). Similarly, Φ(Σ ⃗ 2 ) is tangent to Ξ a−
at a point Φ(p ⃗ − ) and hence ξ ⃗ ∈ G(N ⃗ p− Σ 2 ). Since a − < a + ,
⃗Φ(p − ) ≠ Φ(p ⃗ + ) and then we have proved that ξ ⃗ admits at least
two prei-mages by G. ⃗ (
We claim now that det < G, ⃗ ⃗ )
I > ≥ 0
at these points. Since the whole surface is contained in one of
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is