Conformally Invariant Variational Problems. - SAM

Whence,
[
∆u−H(u)(∇ ⊥ u,∇u) ]·ν(u) = ∆u·ν(u)
= div(∇u·ν(u))−∇u·∇(ν(u)) = −∇u·∇(ν(u))
(VI.28)
where we have used the fact that ∇u · ν(u) = 0 holds almost
everywhere, since ∇u is tangent to N n .
Altogether, (VI.27) and (VI.28) show that u satisfies in the
sense of distributions the equation
∆u−H(u)(∇ ⊥ u,∇u) = −ν(u) ∇(ν(u))·∇u .
(VI.29)
In codimension 1, the second fundamental form acts on a pair
of vectors (U,V) in T z N n via
A z (U,V) = ν(z) < dν z U,V > ,
(VI.30)
so that, as announced, (VI.29) and (VI.25) are identical.
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