Conformally Invariant Variational Problems. - SAM

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Exchangingtheroleofiandj assumingmoreoverthatj ≠ k and subtracting to (II.14) the corresponding identity for the triplet (j,i,k) instead of (i,j,k) gives ∂ 2 e −λ ∂x j ∂x k ∂u − ∂2 e −λ ∂x i ∂x i ∂x k ∂u ∂x j = 0 (II.15) We are assuming that n 3 hence, for any pair i ≠ k there exists a j ≠ k and j ≠ i. Applying (II.15) to this triplet and since ∂u ∂x i and ∂u ∂x j arelinearlyindependentweobtaininparticular ∂ 2 e −λ ∀i ≠ k = 0 (II.16) ∂x i ∂x k We can exchange the role playedby the canonicalbasis withany orthonormal basis of R n . Hence, considering the bilinear form given by the Hessian of e −λ , we have proved that U⊥V =⇒ d 2 e −λ (U,V) = 0 For any U ≠ 0 we introduce L U ∈ (R n ) ∗ given by (II.17) ∀V ∈ R n L U (v) : = d 2 e −λ (U,V)− d2 e −λ (U,U) | U | 2 < U,V > From (II.16) we have that L U is identically zero on U ⊥ and that L U (U) = 0 thus for any U ∈ R n L U ≡ 0 i.e. ∀U ∈ R n ∀V ∈ R n d 2 e −λ (U,V) = d2 e −λ (U,U) | U | 2 < U,V > Arguing as in the end of the proof of Lemma II.3 gives the existence of f ∈ C 1 such that and ∀U,V ∈ R n d 2 e −λ (U,V) = f < U,V > f = ∂2 e −λ ∂x 2 i 10 ∀i = 1...m
Let k ∈ {1,...,n} and choose i ≠ k, using (II.16) ∂f = ∂ ( ∂ 2 e −λ ) = 0 ∂x k ∂x i ∂x i ∂x k f is therefore constant on the domain U and we finally easily deduce the existence of two real constants A and B and a vector x 0 ∈ R n such that e −λ = A | x−x 0 | 2 +B which concludes the proof of the Lemma II.3 ✷ Since U is assumed to be a diffeo- Proof of Theorem II.1 morphism we have taking the differential gives ∀y ∈ u(U) u◦u −1 (y) = y ∀y ∈ u(U) du u −1 (y) ◦du −1 y = I n Where I n is the identity matrix on R n For any pair of vectors Y and Y ′ in R n , using Lemma II.3, we then have denoting x = u −1 (y) < Y,Y ′ > =< du x ·(du −1 y ·Y),du x·(du −1 y Y ′ ) > = 1 (A | x−x 0 | 2 +B) 2 < du−1 y ·Y,du −1 y Y ′ > (II.18) Since u is conformal, u −1 is also conformal and there exists C,D ∈ R and y 0 ∈ R n s.t. ∀y ∈ u(U) < du −1 y Y,du−1 y ′ Y ′ >= 11 1 (C | y −y 0 | 2 +D) 2 < Y,Y ′ > (II.19)
Page 1 and 2: Conformally Invariant Variational P
Page 3 and 4: II Conformal transformations - some
Page 5 and 6: this implies ∀X,Y ∈ R m \{0} if
Page 7 and 8: This form is called the Hopf differ
Page 9: ∂ ∂x k (e 2λ δ ij ) = ∂ ∂
Page 13 and 14: If neither A = 0 nor B = 0 The left
Page 15 and 16: ✷ Proof of Lemma II.4 Since u(0)
Page 17 and 18: III Elementary Differential geometr
Page 19 and 20: control it gives on the image itsel
Page 21 and 22: This is the main advantage of worki
Page 23 and 24: oundary, and sending ∂D 2 monoton
Page 25 and 26: Such a Jordan curve is also simply
Page 27 and 28: Therefore, for m = 3, any minimizer
Page 29 and 30: critical point for variations in th
Page 31 and 32: Remark V.3. There are situations wh
Page 33 and 34: Thus the flow x t of the vector-fie
Page 35 and 36: Another difficulty lies in the rema
Page 37 and 38: and ‖u(x)−u(y)‖ 2 L ∞ ((∂
Page 39 and 40: in the middle of this arc : |p −
Page 41 and 42: V.3 Existence of Parametric disc ex
Page 43 and 44: We further assume that L is conform
Page 45 and 46: We note that Γ i (∇u,∇u) :=
Page 47 and 48: is explicitly given by u(x,y) := lo
Page 49 and 50: Example 3. We consider a map (ω ij
Page 51 and 52: norm. The analytical difficulties r
Page 53 and 54: acting on maps u ∈ W 1,2 (D 2 ,N
Page 55 and 56: Whence, [ ∆u−H(u)(∇ ⊥ u,∇
Page 57 and 58: VII Integrabilitybycompensationtheo
Page 59 and 60: Accordingly, if φ lies in L ∞ ,
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Proof of the regularity of the solu
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If wemultiplytheLaplaceequationthro
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where u still denotes the normal un
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frames, thereby compensating for th
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tangent frame field to T 2 . Define
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egularity. Note that (VI.26) is equ
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Just as in the proof of the regular
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VIII A proof of Heinz-Hildebrandt
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maps, namely ∑n+1 ( −∆u i =
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Theorem VIII.2. [Riv1] Let N n be a
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If A is almost everywhere invertibl
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the whole regularity result stated
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Bringing altogether (VIII.15), (VII
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In the simpler case when Ω is dive
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Theorem VIII.5. [Uhl], [Riv1] Let m
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yields the existence of the solutio
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IX A PDE version of the constant va
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well known variation of the constan
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elliptic estimates give ‖∆ −1
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a meaning to (IX.61) we need at lea
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one, similar approaches can be very
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form associated to g on S at the po
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where g(S) denotes the genus of S.
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- General relativity : The Willmore
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where ⋆ is the Hodge operator 37
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Let us take locally about p a norma
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and that we have denoted by ( ⃗
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and similarly the second fundamenta
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Let (⃗n 1 ,··· ,⃗n m−2 ) b
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orthonormal basis for the metric g.
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X.4.2 Li-Yau Energy lower bounds an
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Thus ⃗G ∗ ω S m−1(p) = 1 |S
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the half space given by the affine
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where we used that the unit sphere
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Conjecture X.1. Let ⃗ Φ be an im
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dle to Φ(Σ ⃗ 2 ) : for all X
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Let TR m ⃗ Φ([0,1]×Σ 2 ) be th
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imply D ∂ ⃗H = 1 ∂t 2 2∑ D
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We have moreover ∇ es ⃗ V N = =
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for any perturbation V ⃗ which is
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the Willmore Lagrangian ”controls
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written in isothermal coordinates.
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quantity div [ 2∇H ⃗ −3H∇
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where e λ = |∂ x1Φ| ⃗ = |∂x
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Observe that ⎧⎪ ⎨ ⎪ ⎩ 〈
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and consequently ⋆(⃗n∧∇ ⊥
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In one hand the projection into the
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The tangential projection gives 4e
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X.6 Construction of Isothermal Coor
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the Lie algebra iR. Sections of thi
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R m is given by ∇ X σ := π T (d
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Let Σ 2 be a smoothcompactoriented
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There exists a constant C > 0, such
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such a way that ⃗n ρ λ realizes
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We make now use of (X.87) and we de
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We are now in positionto start the
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isabilipschitzdiffeomorphismbetween
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Having defined weak Willmore immers
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Let µ be the solution of ⎧ ⎨
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Combining this inequality with (X.1
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ii) iii) iv) limsupArea( Φ ⃗ k (
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The assumption i) implies that, mod
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Theorem X.14. Let ⃗ Φ be a Lipsc
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Thus < ∇ ⃗ Φ,∇ ⊥ ⃗ L >=
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To that purpose we first compute
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and the following equation holds
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One verifies easily that π T ( ⃗
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From (X.197) we compute ⋆(⃗n
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For any such a vector field X satis
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X.7.4 The conformal Willmore surfac
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Denote ∂ z ⃗ L = A⃗ez +B⃗e
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We have using (X.113) ∂ z ∂ z
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References [Ad] Adams, David R. ”
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[DHKW1] Dierkes, Ulrich; Hildebrand
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larityof weak solutionsof nonlinear
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[Poi] Poisson, Siméon Denis ”Ext
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Conformally Invariant Variational Problems. - SAM

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