Conformally Invariant Variational Problems. - SAM

II Conformal transformations - some fundamental
results.
Conformal invariance is a universal property. A transformation
is called conformal when it preserves angles infinitesimally, that
is, when its differential is a similarity at every point. Unlike in
higher dimensions, the group of conformal transformations in
two dimensions is very large ; it has infinite dimension. In fact,
it contains as many elements as there are holomorphic and antiholomorphic
maps. This particularly rich feature motivates us
to restrict our attention on the two-dimensional case. Although
we shall not be concerned with higher dimension, the reader
should know that many of the results presented in these notes
can be generalized to any dimension.
Definition II.1. A C 1 map u between two riemannian manifolds
(M m ,g) and (N n ,h) is said to be conformal if at every
point p ∈ M m du p is a composition of isometries from T p M m
into du p (T p M m ) and dilations. This is equivalent to
∀p ∈ M m ∀X,Y ∈ T p M m
< du p ·X,du p ·Y > h |X| g |Y| g
= |du p ·X| h |du p ·Y| h < X,Y > g
(II.1)
Where < ·,· > g and < ·,· > h denotes respectively the scalar
products g on T p M m and h on du p (T p M) C T u(p) N n . ✷
Lemma II.1. A map u is conformal if and only if there exists
a function λ on M m such that
∀p ∈ M m ∀X,Y ∈ T p M m
(II.2)
< du p ·X,du p ·Y > h = e 2λ(p) < X,Y > g
λ(p) is called the conformal factor at p. ✷
This result comes from the following Lemma in linear algebra.
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