Conformally Invariant Variational Problems. - SAM

ounded second fundamental form as follows :
⎧
⎪⎨
⃗Φ ∈ W 1,∞ (Σ,R m ) s.t. ⃗ ⎫
Φ satisfies (X.144) ⎪⎬
E Σ := ∫
⎪⎩ and |d⃗n| 2 g dvol g < +∞ ⎪⎭
Σ
When Σ is not compact we extend the definition of E Σ as follows
: we require Φ ⃗ ∈ W 1,∞
loc (Σ,Rm ), we require that (X.144) holds
locally on any compact subset of Σ 2 and we still require the
global L 2 control of the second fundamental form
∫
|d⃗n| 2 g dvol g < +∞ .
Σ
X.6.3 Energy controlled liftings of W 1,2 −maps into the Grassman
manifold Gr 2 (R m ).
In the next subsection we will apply the Chern moving frame
methodinthecontextoflipschitzImmersionswithL 2 −bounded
Second Fundamental Form. To that aim we need first to construct
local Coulomb tangent moving frames with controlled
W 1,2 energy. We shall do it in two steps. First we will explore
the possibility to ”lift” the Gauss map and to construct
tangent moving frame with bounded W 1,2 −energy. This is the
purpose of the present subsection. The following result lifting
theorem proved by F.Hélein in [He].
Theorem X.8. Let ⃗n be a W 1,2 map from the disc D 2 into the
Grassman manifold 59 of orientedm−2-planesinR m : Gr m−2 (R m ).
59 The Grassman manifold Gr m−2 (R m ) can be seen as being the sub-manifold of the
euclidian space ∧ m−2 R m of m−2-vectors in R m made of unit simple m−2-vectors and
then one defines
W 1,2 (D 2 ,Gr m−2 (R m )) := { ⃗n ∈ W 1,2 (D 2 ,∧ m−2 R m ) ; ⃗n ∈ Gr m−2 (R m ) a.e. } .
.
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