Conformally Invariant Variational Problems. - SAM

The form H appearing in the Euler-Lagrange equation (VI.15)
is the unique solution of
∀z ∈ R m ∀U,V,W ∈ R m
dω z (U,V,W) = 4U ·H(V,W)
= 4
m∑
U i H i (V,W) .
For instance, in dimension three, dω is a 3-form which can be
identified with a function on R m . More precisely, there exists
H such that dω = 4H dz 1 ∧dz 2 ∧dz 3 . In this notation (VI.15)
may be recast, for each i ∈ {1,...,m}, as
i=1
∆u i = 2H(u) ∂ x u i+1 ∂ y u i−1 −∂ x u i−1 ∂ y u i+1 ,
(VI.16)
where the indexing is understood in Z 3 . The equation (VI.16)
may also be written
∆u = 2H(u) ∂ x u×∂ y u ,
which we recognize as (??), the prescribed mean curvature equation.
In a general fashion, the equation (VI.15) admits the following
geometric interpretation. Let u be a conformal solution of
(VI.15), so that u(D 2 ) is a surface whose mean curvature vector
at the point (x,y) is given by
⎛
⎞
∑ m
e −2λ u ∗ H = ⎝e −2λ Hkl i (u) ∇⊥ u k ·∇u l ⎠ , (VI.17)
k,l=1
i=1···m
where e λ is the conformal factor e λ = |∂ x u| = |∂ y u|. As in
Example 2, the equation (VI.15) forms an elliptic system with
quadratic growth, thus critical in dimension two for the W 1,2
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