Conformally Invariant Variational Problems. - SAM

the Willmore Lagrangian ”controls” the L 2 norm of the
meancurvatureandtheL 2 normofthesecondfundamental
form for a given closed surface for instance. However the
non-linearity in the Eular Lagrange equation in the form
(X.64) is cubic in the second fundamental form. Having
some weak notion of immersionswith only L 2 -bounded second
fundamental form would then be insufficient to write
the equation though the Lagrangian from which this equation
is deduced would make sense for such a weak immersion
! This provides some apparent functional analysis
paradox.
X.5.2 The conservative form of Willmore surfaces equation.
In [Riv2] an alternative form to the Euler Lagrange equation
of Willmore functional was proposed. We the following result
plays a central role in the rest of the course.
Theorem X.7. Let ⃗ Φ be a smooth immersion of a two dimensional
manifold Σ 2 into R m then the following identity holds
= 1 2 d∗ g
∆ ⊥
⃗ H + Ã( ⃗ H)−2| ⃗ H| 2 ⃗ H
[
dH ⃗ −3π ⃗n (dH)+⋆(∗ ⃗ g d⃗n∧ H) ⃗ ] (X.85)
where H ⃗ is the mean curvature vector of the immersion Φ, ⃗ ∆ ⊥
is the negative covariant laplacian on the normal bundle to the
immersion, Ã is the linear map given by (X.65), ∗ g is the Hodge
operator associated to the pull-back metric Φ ⃗ ∗ g R
m on Σ 2 , d ∗ g
=
−∗ g d∗ g is the adjoint operator with respect to the metric g to
the exterior differential d, ⃗n is the Gauss map to the immersion,
π ⃗n is the orthogonal projection onto the normal space to the tangent
space Φ ⃗ ∗ TΣ 2 and ⋆ is the Hodge operator from ∧ p R m into
∧ m−p R m for the canonical metric in R m . ✷
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