Conformally Invariant Variational Problems. - SAM

Note that in the special case when ω = 0, the equation (VI.25)
reduces to
∆u+A(u)(∇u,∇u)= 0 , (VI.26)
which is known as the N n -valued harmonic map equation.
We now establish (VI.25) in the codimension 1 case. Let ν
be the normal unit vector to N. The form ω may be naturally
extended on a small neighborhood of N n via the pull-back πN ∗ ω
of the projection π N . Infinitesimally, to first order, considering
variations for E ω of the form π N (u + tξ) is tantamount to
considering variations of the kind u+tdπ N (u)ξ, which further
amounts to focusing on variations of the form u + tv, where
v ∈ W 1,2 (D 2 ,R m )∩L ∞ satisfies v·ν(u) = 0 almost everywhere.
Following the argument from Example 3, we obtain that u is a
critical point of E ω whenever for all v with v · ν(u) = 0 a.e.,
there holds
⎡
⎤
∫
m∑
⎣∆u i −2 Hkl(u) i ∇ ⊥ u k ·∇u l ⎦ v i dx dy = 0 ,
D 2 m
∑
i=1
k,l=1
where H is the vector-valued two-form on R m given for z on N n
by
∀ U,V,W ∈ R m dπ ∗ Nω(U,V,W) := U ·H z (V,W) .
In the sense of distributions, we thus find that
[
∆u−H(u)(∇ ⊥ u,∇u) ] ∧ν(u) = 0 . (VI.27)
Recall, ν ◦ u ∈ L ∞ ∩ W 1,2 (D 2 ,R m ). Accordingly (VI.27) does
indeed make sense in D ′ (D 2 ).
Note that if any of the vectors U, V, and W is normal to N n ,
i.e. parallel to ν, then dπN ∗ ω(U,V,W) = 0, so that
ν z ·H z (V,W) = 0 ∀ V ,W ∈ R m .
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