Conformally Invariant Variational Problems. - SAM

form associated to g on S at the point p is given by
√
dvol g := det(g(∂ xi ,∂ xj )) dx 1 ∧dx 2 ,
where (x 1 ,x 2 ) are arbitrary local positive coordinates 31
The second fundamental form at p ∈ S is the bilinear map
which assigns to a pair of vectors ⃗ X, ⃗ Y in T p S an orthogonal
vector to T p S that we shall denote ⃗ I( ⃗ X, ⃗ Y). This normal vector
”expresses” how much the Gauss map varies along these directions
⃗ X and ⃗ Y. Precisely it is given by
⃗ Ip : T p S ×T p S −→ N p S
( X, ⃗ Y) ⃗ 〈
−→ − d⃗n p · ⃗X, Y ⃗ 〉
⃗n(p)
(X.1)
Extending smoothly X ⃗ and Y ⃗ locally first on S and then in a
neighborhood of p in R 3 , since < ⃗n,Y >= 0 on S one has
⃗ Ip ( X, ⃗ Y) ⃗ 〈
:= ⃗n p·,dY ⃗ p · ⃗X
〉
⃗n = dY ⃗ p ·X −∇ ⃗X Y ⃗
(X.2)
= ∇ ⃗X Y ⃗ −∇⃗X Y ⃗ .
where ∇ is the Levi-Civita connection on S generated by g, it is
given by π T (d ⃗ Y p·X) where π T is the orthogonal projection onto
T p S, and ∇ is the the Levi-Civita connection associated to the
flat metric and is simply given by ∇ ⃗X
⃗ Y = d ⃗ Yp ·X.
An elementary but fundamental property of the second fundamental
form says that it is symmetric 32 . It can then be diago-
31 Local coordinates, denoted (x 1 ,x 2 ) is a diffeomorphism x from an open set in R 2 into
an open set in Σ. For any point q in this open set of S we shall denote x i (q) the canonical
coordinates in R 2 of x −1 (q). Finally ∂ xi is the vector-field on S given by ∂x/∂x i .
32 This can be seen combining equation (X.2) with the fact that the two Levi Civita
connections ∇ and ∇ are symmetric and hence we have respectively
∇ ⃗X
⃗ Y −∇⃗Y ⃗ X = [X,Y]
and
∇ ⃗X
⃗ Y −∇⃗Y ⃗ X = [X,Y] .
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