Conformally Invariant Variational Problems. - SAM

where ⃗ Φ runs over all conformal immersions of (Σ 2 ,c).
Let Φ ⃗ be a conformal immersion of a surface Σ 2 into R m and
π be the inverse of the stereographic projection from R m into
S m (which is a conformal map). Corollary X.1 gives
∫
W( Φ) ⃗ = | H
∫Σ ⃗ π◦Φ ⃗| 2 dvol (π◦Φ)∗ ⃗ g + S
dvol 2 m (π◦Φ)∗ ⃗ g S
.
Σ 2 m
from which one deduces the following lemma.
Lemma X.1. Let (Σ 2 ,c) be a closed Riemann surface and ⃗ Φ
be a conformal immersion of this surface. Then the following
inequality holds
∫
Σ 2 | ⃗ H ⃗Φ | 2 dvol ⃗Φ∗ g R m ≥ V c(m,Σ 2 ) .
Moreover equality holds if and only if Φ(Σ ⃗ 2 ) is the stereographic
projection of a minimal surface of S m . ✷
The main achievement of [LiYa] is to provide lower bounds
of V c (m,Σ 2 ) in terms of the conformal class of Σ 2 . In particular
they establish the following result
Theorem X.4. Let (T 2 ,c) be a torus equipped with the conformal
class given by the flat torus R 2 /aZ + bZ where a = (1,0)
and b = (x,y) where 0 ≤ x ≤ 1/2 and √ 1−x 2 ≤ y ≤ 1, then
for any m ≥ 3
2π 2 ≤ V c (m,(T 2 ,c)) .
Combining lemma X.1 and theorem X.4 gives 2π 2 as a lower
bound to the Willmore energy of conformal immersions of riemann
surfaces in some sub-domain of Moduli space of the torus.
InfactthefollowingstatementhasbeenconjecturedbyT.Willmore
in 1965
128
✷