Conformally Invariant Variational Problems. - SAM

Theorem VIII.2. [Riv1] Let N n be an arbitrary closed oriented
C 2 -submanifold of R m , with 1 ≤ n < m, and let ω be a C 1 twoform
on N n . Suppose that u is a critical point in W 1,2 (D 2 ,N n )
of the energy
E ω (u) = 1 ∫
|∇u| 2 (x,y) dx dy +u ∗ ω .
2 D 2
Then u fulfills all of the hypotheses of theoreme VIII.1, and
therefore is Hölder continuous.
✷
Proof of theorem VIII.2.
The critical points of E ω satisfy the equation (VI.25), which,
in local coordinates, takes the form
∆u i = −
m∑
j,k=1
H i jk (u) ∇⊥ u k ·∇u j −
m∑
A i jk (u) ∇uk ·∇u j ,
j,k=1
(VIII.7)
for i = 1···m. Denoting by (ε i ) i=1···m the canonicalbasis of R m ,
we first observe that since
H i jk (z) = dω z(ε i ,ε j ε k )
the antisymmetry of the 3-forme dω yields for every z ∈ R m the
identity Hjk i (z) = −Hj ik
(z). Then, (VIII.7) becomes
∆u i = −
m∑
(Hjk i (u)−Hj ik (u))∇⊥ u k·∇u j −
j,k=1
m∑
j,k=1
A i jk (u)∇uk·∇u j .
(VIII.8)
On the other hand, A(u)(U,V) is orthogonal to the tangent
plane for every choice of vectors U et V 22 . In particular, there
22 Rigorously speaking, A is only defined for pairs of vectors which are tangent to the
surface. Nevertheless, A can be extended to all pairs of vectors in R m in a neighborhood
of N n by applying the pull-back of the projection on N n . This extension procedure is
analogous to that outlined on page 18.
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