Conformally Invariant Variational Problems. - SAM

Note that Example 1 is contained as a particular case.
Verifying that E g is indeed conformally invariant may be done
analogously to the case of the Dirichlet energy, via introducing
the complex variable z = x+iy. No new difficulty arises, and
the details are left to the reader as an exercise.
TheweakcriticalpointsofE g arethefunctionsu ∈ W 1,2 (D 2 ,R m )
which satisfy
∀ξ ∈ C ∞ 0 (D 2 ,R m )
d
dt E g(u+tξ) |t=0 = 0 .
An elementary computation reveals that u is a weak critical
point of E g if and only if the following Euler-Lagrange equation
holds in the sense of distributions:
m∑
∀i = 1···m ∆u i + Γ i kl (u)∇uk ·∇u l = 0 . (VI.4)
k,l=1
Here, Γ i kl aretheChristoffelsymbolscorrespondingtothemetric
g, explicitly given by
Γ i kl(z) = 1 2
m∑
g is (∂ zl g km +∂ zk g lm −∂ zm g kl ) ,
s=1
(VI.5)
where (g ij ) is the inverse matrix of (g ij ).
Equation (VI.4) bears the name harmonic map equation 6
with values in (R m ,g).
Just as in the flat setting, if we further suppose that u is
conformal, then (VI.4) is in fact equivalent to u(D 2 ) being a
minimal surface in (R m ,g).
6 One wayto interpret(VI.4) asthe two-dimensionalequivalent ofthe geodesicequation
in normal parametrization,
d 2 x i
dt 2 + m ∑
k,l=1
Γ i dx k
kl
dt
dx l
dt = 0 .
44