Conformally Invariant Variational Problems. - SAM

Conjecture X.1. Let ⃗ Φ be an immersion in R m of the two
dimensional torus T 2 then
∫
T 2 | ⃗ H ⃗Φ | 2 dvol ⃗Φ∗ g R m ≥ 2π2
Equality should hold only for Φ(T 2 ) being equal to a Moebius
transform of the stereographic projection into R 3 of the Clifford
torus Tcliff 2 := {1/√ 2 (e iθ ,e iφ ) ∈ C 2 ;(θ,φ) ∈ R 2 } ⊂ R 3 . ✷
This conjecture has stimulated a lot of works. Recently F.C.
Marques and A.Neves have submitted a proof of it in codimension
1 : m = 3 (see [MaNe]).
X.5 The Willmore Surface Equations.
In the previoussubsection we havepresented some lowerbounds
for the Willmore energy under various constraints. It is natural
to look at the existence of the optimal surfaces for which
the Willmore energy achieves it’s lower bound. For instance
minimal surfaces S are absolute minima of the Willmore energy
since they satisfy ⃗ H = 0 and then W(S) = 0. More generally
theseoptimalimmersedsurfacesundervariousconstraints(fixed
genus, boundary values...etc) will be critical points to the Willmore
energy that are called Willmore surfaces. As we saw the
Willmore energy in conformal coordinates identify with the L 2
norm of the immersion. It is therefore a 4-th order PDE generalizing
the minimal surface equation ⃗ H ⃗Φ = 0 which is of second
order 45 (This is reminiscent for instance to the bi-harmonic map
equation which is the 4th order generalization of the harmonic
map equation which is of order 2 or, in the linear world, the
Laplace equation being the 2nd order version of the Cauchy-
Riemann which is a 1st order PDE ...etc). This idea of having
a 4-th order generalization of the minimal surface equation was
45 Recall that in conformal coordinates ⃗ H ⃗Φ = 0 in R m is equivalent to ∆ ⃗ Φ = 0.
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