Conformally Invariant Variational Problems. - SAM

have a smooth minimizer u Γ of the Dirichlet energy E among
ourspaceofC 1 immersionssendinghomeomorphically∂D 2 onto
the given Jordan curve Γ, then we claim that u Γ is conformal
and minimizes also A in the class. Indeed first, if u Γ would not
minimizeA in thisclass, there wouldthen be anotherimmersion
v such that
A(v) < E(u Γ ) .
Let g be pull-back metric on the disc D 2 , g := v ∗ g R m, where g R
m
denotes the standard scalar product in R m . The uniformization
theoremgivestheexistenceofadiffeomorphismΨinDiff + (D 2 )
such that Ψ ∗ g is conformal :
e 2λ [dx 2 1 +dx2 2 ] = Ψ∗ g = Ψ ∗ v ∗ g R
m = (v ◦Ψ) ∗ g R
m .
In other words we have that v ◦ Ψ is conformal therefore the
following strict inequality holds
E(v ◦Ψ) = A(v ◦Ψ) = A(v) < E(u Γ ) ,
which is a contradiction.
For similar reasons u Γ has to be conformal. Indeed if this
would not be the case we would again find a diffeomorphism Ψ
such that u Γ ◦Ψ is conformal and, under this assumption that
u Γ is not conformal we would have
E(u Γ ◦Ψ) = A(u Γ ◦Ψ) = A(u Γ ) < E(u Γ ) ,
which would be again a contradiction.
Of course this heuristic argument is based on the hypothesis
that we have found a minimizer and that this minimizer is a
smooth immersion, since we applied the uniformizationtheorem
to the induced metric u ∗ Γ g Rm. In order to use the ”nice” functional
properties of the lagrangianE, as we mentioned above we
have to enlarge the class of candidates for the minimization to
the space of Sobolev maps in W 1,2 (D 2 ,R m ), continuous at the
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