Conformally Invariant Variational Problems. - SAM

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iemannian manifold (M m ,g). Let Ψ be a positive conformal diffeomorphism from (M m ,g) into another riemannian oriented manifold (N m ,k) then we have the following pointwise identity everywhere on Σ 2 [ | H ⃗ Φ ⃗ ∗g | 2 Φ −K ⃗ ∗g +K g] dvol ⃗Φ∗ g [ = | H ⃗ (Ψ◦⃗ Φ) ∗k | 2 −K (Ψ◦⃗ Φ) ∗k +K k] dvol (Ψ◦Φ)∗ ⃗ k (X.47) where K g (resp. K k ) is the sectional curvature of the 2−plane ⃗Φ ∗ TΣ 2 in (M m ,g) (resp. of the two plane Ψ ∗Φ∗ ⃗ TΣ 2 in (N m ,k)). In particular the following equality holds ∫ W( Φ)+ ⃗ K g dvol ⃗Φ∗ g Σ 2 ∫ (X.48) = W(Ψ◦Φ)+ ⃗ K k dvol (Ψ◦Φ)∗ ⃗ g . Σ 2 Proof of corollary X.1. By definition Ψ realizes an isometry between (M m ,Ψ ∗ k) and (N m ,k). Let µ ∈ R such that e µ g = Ψ ∗ k. We can apply the previous theorem with h = Ψ ∗ k and we obtain [ | ⃗ H ⃗ Φ ∗g | 2 −K ⃗ Φ ∗ g ] = e 2µ [ | ⃗ H (Ψ◦⃗ Φ) ∗k | 2 −K (Ψ◦⃗ Φ) ∗ k ] ✷ . (X.49) Itisalsoclearthatthevolumeformdvol g givenbytherestriction to Σ 2 of the metric g is equal to e −2µ dvol Ψ∗ k where dvol Ψ∗ k is equal to the volume form given by the restriction to Σ 2 of the metric Ψ ∗ k. Hence combining this last fact with (X.47). (X.48) is obtained by integrating (X.47) over Σ 2 , the scalar curvature terms canceling each other on both sides of the identity due to Gauss Bonnet theorem. Corollary X.1 is then proved. ✷ 120
X.4.2 Li-Yau Energy lower bounds and the Willmore conjecture. It is a well known fact that the integral of the curvature κ ⃗Φ of the immersion Φ ⃗ of a closed curved Γ (∂Γ = ∅) in R m is always larger than 2π : ∫ F( Φ) ⃗ = κ ⃗Φ ≥ 2π , (X.50) Γ and is equal to 2π if and only if Γ ≃ S 1 and it’s image by Φ ⃗ realizes a convex planar curve. Using the computations in the previous section one easily verifies that F is conformally invariant F(Ψ◦Φ) ⃗ = F( Φ) ⃗ and plays a bit the role of a 1-dimensional version of the Willmore functional. As the immersion gets more complicated one can expect the lower bound (X.50) to increase. For instance a result by Milnor says that if the immersion Φ ⃗ is a knotted embedding then the lower bound (X.50) is multiplied by 2 : ∫ κ ⃗Φ ≥ 4π . (X.51) Γ It is natural to think that this general philosophy can be transfered to the Willmore functional : if the surface Σ 2 and the immersion ⃗ Φ become ”more complicated” then there should exist increasing general lower bounds for W( ⃗ Φ). First we give the result corresponding to the first lower bound (X.50) for curves. It wasfirst proved by ThomasWillmorefor m = 3 and by Bang- Yen Chen [BYC2] in the general codimension case. Theorem X.2. Let Σ 2 be a closed surface and let Φ ⃗ be a smooth immersion of Σ 2 into R m . Then the following inequality holds ∫ W( Φ) ⃗ = | H ⃗ ⃗Φ | 2 dvol ⃗Φ∗ g ≥ 4π . (X.52) R Σ 2 m Moreover equality holds if and only if Σ 2 is a 2-sphere and Φ ⃗ realizes - modulo translations and dilations - an embedding onto S 2 the unit sphere of R 3 ⊂ R m . ✷ 121
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Conformally Invariant Variational P
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II Conformal transformations - some
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this implies ∀X,Y ∈ R m \{0} if
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This form is called the Hopf differ
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∂ ∂x k (e 2λ δ ij ) = ∂ ∂
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Let k ∈ {1,...,n} and choose i
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If neither A = 0 nor B = 0 The left
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✷ Proof of Lemma II.4 Since u(0)
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III Elementary Differential geometr
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control it gives on the image itsel
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This is the main advantage of worki
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oundary, and sending ∂D 2 monoton
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Such a Jordan curve is also simply
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Therefore, for m = 3, any minimizer
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critical point for variations in th
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Remark V.3. There are situations wh
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Thus the flow x t of the vector-fie
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Another difficulty lies in the rema
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and ‖u(x)−u(y)‖ 2 L ∞ ((∂
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in the middle of this arc : |p −
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V.3 Existence of Parametric disc ex
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We further assume that L is conform
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We note that Γ i (∇u,∇u) :=
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is explicitly given by u(x,y) := lo
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Example 3. We consider a map (ω ij
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norm. The analytical difficulties r
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acting on maps u ∈ W 1,2 (D 2 ,N
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Whence, [ ∆u−H(u)(∇ ⊥ u,∇
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VII Integrabilitybycompensationtheo
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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
Page 69 and 70: tangent frame field to T 2 . Define
Page 71 and 72: egularity. Note that (VI.26) is equ
Page 73 and 74: Just as in the proof of the regular
Page 75 and 76: VIII A proof of Heinz-Hildebrandt
Page 77 and 78: maps, namely ∑n+1 ( −∆u i =
Page 79 and 80: Theorem VIII.2. [Riv1] Let N n be a
Page 81 and 82: If A is almost everywhere invertibl
Page 83 and 84: the whole regularity result stated
Page 85 and 86: Bringing altogether (VIII.15), (VII
Page 87 and 88: In the simpler case when Ω is dive
Page 89 and 90: Theorem VIII.5. [Uhl], [Riv1] Let m
Page 91 and 92: yields the existence of the solutio
Page 93 and 94: IX A PDE version of the constant va
Page 95 and 96: well known variation of the constan
Page 97 and 98: elliptic estimates give ‖∆ −1
Page 99 and 100: a meaning to (IX.61) we need at lea
Page 101 and 102: one, similar approaches can be very
Page 103 and 104: form associated to g on S at the po
Page 105 and 106: where g(S) denotes the genus of S.
Page 107 and 108: - General relativity : The Willmore
Page 109 and 110: where ⋆ is the Hodge operator 37
Page 111 and 112: Let us take locally about p a norma
Page 113 and 114: and that we have denoted by ( ⃗
Page 115 and 116: and similarly the second fundamenta
Page 117 and 118: Let (⃗n 1 ,··· ,⃗n m−2 ) b
Page 119: orthonormal basis for the metric g.
Page 123 and 124: Thus ⃗G ∗ ω S m−1(p) = 1 |S
Page 125 and 126: the half space given by the affine
Page 127 and 128: where we used that the unit sphere
Page 129 and 130: Conjecture X.1. Let ⃗ Φ be an im
Page 131 and 132: dle to Φ(Σ ⃗ 2 ) : for all X
Page 133 and 134: Let TR m ⃗ Φ([0,1]×Σ 2 ) be th
Page 135 and 136: imply D ∂ ⃗H = 1 ∂t 2 2∑ D
Page 137 and 138: We have moreover ∇ es ⃗ V N = =
Page 139 and 140: for any perturbation V ⃗ which is
Page 141 and 142: the Willmore Lagrangian ”controls
Page 143 and 144: written in isothermal coordinates.
Page 145 and 146: quantity div [ 2∇H ⃗ −3H∇
Page 147 and 148: where e λ = |∂ x1Φ| ⃗ = |∂x
Page 149 and 150: Observe that ⎧⎪ ⎨ ⎪ ⎩ 〈
Page 151 and 152: and consequently ⋆(⃗n∧∇ ⊥
Page 153 and 154: In one hand the projection into the
Page 155 and 156: The tangential projection gives 4e
Page 157 and 158: X.6 Construction of Isothermal Coor
Page 159 and 160: the Lie algebra iR. Sections of thi
Page 161 and 162: R m is given by ∇ X σ := π T (d
Page 163 and 164: Let Σ 2 be a smoothcompactoriented
Page 165 and 166: There exists a constant C > 0, such
Page 167 and 168: such a way that ⃗n ρ λ realizes
Page 169 and 170: 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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