Conformally Invariant Variational Problems. - SAM

X.6 Construction of Isothermal Coordinates.
In the previous subsection we discussed the difficulty to work
with the Willmore surfaces equation due to the huge invariance
group given by the space of positive diffeomorphisms of the surface.
A classical way to by-pass this difficulty consists in ”breaking”
the symmetry group (or ”gauge group”) of coordinates by
restricting to a special subclass satisfying the Coulomb condition.
In the present subsection we will explain why this choice
correspondstotheconformalcondition. Wehaveseenintheprevious
subsection that, in such coordinates, the Willmore surface
equations can be written in a conservative-elliptic form which is
critical with respect to the L 2 −norm of the second fundamental
form. This triggers the hope to give answers to the analysis
questions we raised for Willmore surfaces.
Breakingthesymmetrygroupofcoordinatesbytakingisothermal
ones is however not enough per se. The uniformization theorem
tells us that, taking the conformal class defined by the
pull-back metric Φ ⃗ ∗ g R
m on Σ 2 , there is a system of coordinates
on the fundamental domain of either C∪∞, C or the Poincaré
half-plane corresponding to this class in which our immersion is
conformal. However, in a minimization procedure for instance,
taking a minimizing sequence of immersions Φ ⃗ k , assuming the
conformal class defined by Φ ⃗ k on Σ 2 is controlled - is not convergingtotheboundaryofthemodulispace-thereisa-priorino
control of the conformal factor corresponding to the pull-back
metric Φ ⃗ ∗ k g Rm and the fact that we are in conformal coordinates
is not helping much. We need then to have
Conformal coordinates + estimates of the conformal factor.
InGaugetheorysuchasinYang-Millsproblemin4dimension
for instance, the Coulomb choice of gauge is the one that provides
estimates of the connection that will be controlled by the
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