Conformally Invariant Variational Problems. - SAM

The direct minimization of L is here also made difficult by the
existence of a huge non compact group of invariance for L : the
group of positive diffeomorphisms of the segment [0,1] : for any
C 1 function t(s) satisfying t ′ > 0, t(0) = 0 and t(1) = 1 one has
L(γ ◦s) = L(γ) .
The classical strategy to remedy to this difficulty is to minimize
instead the energy of the immersion γ
E(γ) :=
∫ 1
0
|˙γ| 2 g dt ,
in the space W 1,2 ([0,1],M m ). The twolagrangiansbeing related
by the following inequality for any immersion γ
∫
L(γ) =
[0,1]
[∫ 1
] 1/2
|˙γ| dt ≤ |˙γ| 2 g dt
with equality if and only if the curve is in normal parametrisation
:
|˙γ| g = cte a.e
A classical results asserts that the minimization of E provides a
minimizer of L in normal parametrization.
Back to the two dimensional situation, in an ideal scenario,
one could then hope a symmetry between the one dimensional
and the twodimensionalcases in which, normalparametrization
is replaced by conformal one, and to obtain, by minimizing E,
a minimizer of A in conformal parametrization. This is eventuallywhatwillhappen
atthe end but the2-dimensionalsituation
is analytically more complex and requires more work to be described
as we will see below.
In one dimension the Dirichlet energy E is invariant under a
finitedimensionalgroup: theconstantspeed reparametrization.
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