Conformally Invariant Variational Problems. - SAM

the Lie algebra iR. Sections of this bundles are given by maps
( ⃗ f 1 , ⃗ f 2 ) from D 2 into ⃗ Φ ∗ (TD 2 × TD 2 ) such that ( ⃗ f 1 , ⃗ f 2 )(x 1 ,x 2 )
realizes a positive orthonormal basis of ⃗ Φ ∗ (T (x1 ,x 2 )D 2 ).
Thepassagefromonesection(⃗e 1 ,⃗e 2 )toanothersection( ⃗ f 1 , ⃗ f 2 )
is realized through a change of gauge which corresponds to the
actionofanSO(2)rotatione iθ onthetangentspace ⃗ Φ ∗ (T (x1 ,x 2 )D 2 )
:
⃗f 1 +i ⃗ f 2 = e iθ (⃗e 1 +i⃗e 2 ) .
The expression of the same connection but in the new trivialization
given by the section ( f ⃗ 1 , f ⃗ 2 ) satisfies the classical gauge
change formula for an S 1 -bundle :
〈
i ⃗f1 ,df ⃗ 〉
2 = i〈⃗e 1 ,d⃗e 2 〉+idθ . (X.130)
The curvature of this connection is given by
id〈⃗e 1 ,d⃗e 2 〉
= i[< D e1 ⃗e 1 ,D e2 ⃗e 2 > − < D e2 ⃗e 1 ,D e1 ⃗e 2 >] e ∗ 1 ∧e∗ 2
[
= i < ⃗ I(⃗e 1 ,⃗e 1 ), ⃗ I(⃗e 2 ,⃗e 2 ) > −| ⃗ I(⃗e 1 ,⃗e 2 )| 2] (X.131)
e ∗ 1 ∧e ∗ 2
= i K dvol g
where we recall the notations we already introduced : e i is the
vector field on D 2 given by d ⃗ Φ · e i = ⃗e i , D ei ⃗e j := π ⃗n (d⃗e j · e i )
and K is the Gauss curvature of (D 2 , ⃗ Φ ∗ g). In the last identity
we have made use of Gauss theorem (theorem 2.5 chap. 6 in
[doC2]).
Combining (X.127) and (X.131) gives the well known expression
of the Gauss curvature in isothermal coordinates in terms
of the conformal factor λ :
−∆λ =< ∇ ⊥ ⃗e 1 ,∇⃗e 2 >= e 2λ K .
(X.132)
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