Conformally Invariant Variational Problems. - SAM

acting on maps u ∈ W 1,2 (D 2 ,N n ). The critical points of E ω are
defined as follows. Let π N be the orthogonal projection on N n
whichtoeachpointinaneighborhoodofN associatesitsnearest
orthogonal projection on N n . For points sufficiently close to N,
the map π N is regular. We decree that u ∈ W 1,2 (D 2 ,N n ) is a
critical point of E ω whenever there holds
d
dt Eω (π N (u+tξ)) t=0 = 0 ,
(VI.24)
for all ξ ∈ C ∞ 0 (D 2 ,R m ).
It can be shown 11 that (VI.24) is satisfied by u ∈ C ∞ 0 (D2 ,R m )
if and only if u obeys the Euler-Lagrange equation
∆u+A(u)(∇u,∇u)= H(u)(∇ ⊥ u,∇u) ,
(VI.25)
where A(≡ A z ) is the second fundamental form at the point
z ∈ N n correspondingtotheimmersionofN n intoR m . Toapair
of vectors in T z N n , the map A z associates a vector orthogonal
to T z N n . In particular, at a point (x,y) ∈ D 2 , the quantity
A (x,y) (u)(∇u,∇u) is the vector of R m given by
A (x,y) (u)(∇u,∇u):= A (x,y) (u)(∂ x u,∂ x u)+A (x,y) (u)(∂ y u,∂ y u) .
For notational convenience, we henceforth omit the subscript
(x,y).
Similarly, H(u)(∇ ⊥ u,∇u) at the point (x,y) ∈ D 2 is the vector
in R m given by
H(u)(∇ ⊥ u,∇u) := H(u)(∂ x u,∂ y u)−H(u)(∂ y u,∂ x u)
= 2H(u)(∂ x u,∂ y u) ,
where H(≡ H z ) is the T z N n -valued alternating two-form on
T z N n :
∀ U,V,W ∈ T z N n dω(U,V,W) := U ·H z (V,W) .
11 in codimension 1, this is done below.
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