Conformally Invariant Variational Problems. - SAM

where we used that the unit sphere is a constant sectional curvature
space : for any two plane σ in TS m K Sm (σ) = 1. In
the other hand the previous discussion leads to the following
inequality
∫
liminf | H
λ→+∞
∫Σ ⃗ Ψλ ◦Φ ⃗|2 dvol (Ψλ ◦Φ) ⃗ ∗ g + S
dvol 2 m (Ψλ ◦Φ) ⃗ ∗ g S
Σ 2 m
k∑
∫
≥ | H| ⃗ 2 g Sj
dvol gSj +Area(S j ) .
j=1 S j
(X.61)
For each S j there is an isometry of S m sending S j to the canonical
2-sphere S 2 ⊂ R 3 ⊂ R m . S 2 is minimal in S m ( H ⃗ ≡ 0 ) and
Area(S 2 ) = 4π. Thus we deduce that
∫
∀j = 1···k | H| ⃗ 2 g Sj
dvol gSj +Area(S j ) = 4π . (X.62)
S j
Combining (X.60), (X.61) and (X.62) gives Li-Yau inequality
(X.59).
✷
Li and Yau established moreover a connection between the
Willmore energy of an immersed surface and it’s the conformal
class that provided new lower bounds.
Let Φ ⃗ be a conformal parametrization of the immersion of a
riemann surface 44 (Σ 2 ,c) into S m . The m−conformal volume
of Φ ⃗ is the following quantity
∫
V c (m, Φ) ⃗ = sup dvol ⃗Φ∗ Ψ ∗ g
Ψ∈Conf(S m S
.
) Σ 2 m
whereConf(S m )denotesthespaceofconformaldiffeomorphism
of S m . Then Li and Yau define the m−conformal volume of a
Rieman surface (Σ 2 ,c) to be the following quantity
V c (m,(Σ 2 ,c)) = inf
⃗Φ
V c (m, ⃗ Φ)
44 The pair (Σ 2 ,c) denotes a closed 2-dimensional manifold Σ 2 together with a fixed
conformal class on this surface.
127