Conformally Invariant Variational Problems. - SAM

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Proof of theorem X.2 Denote by NΣ 2 the pullback by Φ ⃗ of the normal sphere bundle to Φ(Σ ⃗ 2 ) in R m made of the unit normal vectors to Φ ⃗ ∗ TΣ 2 . Denote by G ⃗ = (G 1 ,··· ,G m ) the map from NΣ 2 into S m−1 the unit sphere which to an element in NΣ 2 assigns the corresponding unit vector in S m−1 . For any k ∈ N, k ≥ 1, we denote Observe that ω S k−1 = 1 |S k−1 | ∫ k∑ (−1) j−1 x j ∧ l≠j dx l . j=1 S k−1 ω S k−1 = 1 . Locally on Σ 2 we can choose an orthonormal positively oriented frame of the normal plane : (⃗n 1···⃗n m−2 ). We also chose locally an orthonormal tangent frame (⃗e 1 ,⃗e 2 ). This normal frame permits locally, over an open disk U ⊂ Σ 2 , to trivialize NΣ 2 and the map ⃗ G can be seen as a map from U ×S m−3 into S m−1 : ⃗G(p,s) = m−2 ∑ α=1 s α ⃗n α (p) , where ∑ α s2 α = 1. Let p ∈ U, in order to simplify the notations we may assume that (⃗e 1 ,⃗e 2 ,⃗n 1···⃗n m−2 ) at p coincides with the canonical basis of R m . Hence we have G 1 (p) = G 2 (p) = 0 and G α+2 (p) = s α for α = 1···m−2. and dG 1 (p) = dG 2 (p) = m−2 ∑ α=1 m−2 ∑ α=1 ∀ α = 1···m−2 dG α+2 = ds α + s α < d⃗n α ,⃗e 1 > , s α < d⃗n α ,⃗e 2 > , 122 m−2 ∑ β=1 s β < d⃗n β ,⃗n α > .
Thus ⃗G ∗ ω S m−1(p) = 1 |S m−1 | Observe that m−2 ∧ m−2 ∑ α,β=1 ∑ (−1) j+1 s j ∧ l≠j ds l j=1 = |Sm−3 | |S m−1 | < d⃗e 1 , ⃗ G > ∧ < d⃗e 2 , ⃗ G >= det s α s β < d⃗n α ,⃗e 1 > ∧ < d⃗n α ,⃗e 2 > < d⃗e 1 , ⃗ G > ∧ < d⃗e 2 , ⃗ G > ∧ω S m−3(s) ( < G, ⃗ ⃗ ) I > dvol ⃗Φ∗ g R m (X.53) (X.54) anddenotingΩtheclosedm−3−formonNΣ 2 whichisinvariant under the actionof SO(m−2) overthe fibers and whose integral over each fiber is equal to one 40 , combining (X.53) and (X.54), we have proved finally that 41 ⃗G ∗ ω S m−1 = m−2 ( 2π det < G, ⃗ ⃗ ) I > dvol ⃗Φ∗ g ∧Ω (X.55) R m The form < G, ⃗ ⃗ I > is bilinear symmetric, if κ 1 , κ 2 are it’s eigenvalues, since 4κ 1 κ 2 ≤ (κ 1 + κ 2 ) 2 , we deduce the pointwise inequality ⃗G ∗ ω S m−1 ≤ m−2 tr g (< G, ∣ ⃗ ⃗ I >) 2 ∣ dvolΦ ⃗ ∗ g ∧Ω R m = m−2 2π 2π ∣ 2 ∣ ∣ ∣< G, ⃗ H ⃗ ⃗Φ > ∣ 2 dvol ⃗Φ∗ g ∧Ω R m (X.56) 40 This means that Ω is the Thom class of the normal bundle NΣ 2 it coincides in particular with ω S m−3 in the coordinates s. 41 where we are using the fact that |S m−3 |/|S m−1 | = (m−2)/2π. 123
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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
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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 and 120: orthonormal basis for the metric g.
Page 121: X.4.2 Li-Yau Energy lower bounds an
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
Page 171 and 172: 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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