Conformally Invariant Variational Problems. - SAM

Theorem X.3. Let Σ 2 be a closed surface and let Φ ⃗ be a smooth
immersion of Σ 2 into R m . Assume there exists a point p ∈ R m
with at least k pre-images by Φ, ⃗ then the following inequality
holds
∫
W( Φ) ⃗ = | H ⃗ ⃗Φ | 2 dvol ⃗Φ∗ g ≥ 4πk . (X.59)
R
Σ 2 m
An important corollary of the previous theorem is the following
Li-Yau 8π−threshold result 42 .
Corollary X.2. Let Σ 2 be a closed surface and let Φ ⃗ be a smooth
immersion of Σ 2 into R m . If
∫
W( Φ) ⃗ = | H ⃗ ⃗Φ | 2 dvol ⃗Φ∗ g < 8π ,
R
Σ 2 m
then ⃗ Φ is an embedding 43 .
Proof of theorem X.5. Assume the origin O of R m admits
k−pre-images by Φ. ⃗ Let π be inverse of the stereographic projection
of R m into S m which sends O to the north pole of S m
and ∞ to the south pole. Let D λ be the homothecy of center
O and factor λ. On every ball B R (O), the restriction of D λ ◦Φ
⃗
converge in any C l norm to a union of k planes restricted to
B R (O) as λ goes to +∞. Hence π◦D λ ◦Φ(Σ ⃗ 2 ) restricted to any
compact subset of S m \ {southpole} converges strongly in any
C l norm to a union of totally geodesic 2-spheres S 1 ,··· ,S k . It
is well known that Ψ λ := π◦D λ are conformal transformations.
Applying then corollary X.1 we have in one hand
∫
W( Φ) ⃗ = | H
∫Σ ⃗ Ψλ ◦Φ ⃗|2 dvol (Ψλ ◦Φ) ⃗ ∗ g + S
dvol 2 m (Ψλ ◦Φ) ⃗ ∗ g S
.
Σ 2 m
✷
✷
(X.60)
42 A weaker version of this result has been first established for spheres in codimension 2
(i.e. m = 4) by Peter Wintgen [Win].
43 We recall that embeddings are injective immersions. Images of manifolds by embeddings
are then submanifolds.
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