Conformally Invariant Variational Problems. - SAM

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A straightforwardbut importantconsequence of theoremX.7 is the following conservative form of Willmore surfaces equations. Corollary X.3. An immersion Φ ⃗ of a 2-dimensional manifold Σ 2 is Willmore if and only if the 1-form given by ∗ g [dH ⃗ −3π ⃗n (dH)+⋆(∗ ⃗ g d⃗n∧ H) ⃗ ] is closed. ✷ The analysis questions for Willmore immersions we raised in the previous subsection can be studied basically from two point of views i) By working with the maps ⃗ Φ themseves. ii) By workingwiththe immersedsurface: theimage ⃗ Φ(Σ 2 ) ⊂ R m . The drawback of the first approach is the huge invariance group of the problem containing the positive diffeomorphism group of Σ 2 . The drawback of the second approach comes from the fact that the Euler Lagrange equation is defined on an unknown object ⃗ Φ(Σ 2 ). In thiscourse we shalltake the first approachbut by trying to ”break” as much as we can the symmetry invariance given by the action of positive diffeomorphisms of Σ 2 . From Gauge theory and in particular from Yang-Mills theory a natural way to ”break” a symmetry group is to look for Coulomb gauges. As we will explain the concept of Coulomb gauge transposed to the present setting of immersions of 2-dimensional manifolds is given by the Isothermal coordinates (or conformal parametrization). We shall make an intensive use of these conformal parametrization and therefore we shall make an intensive use of the conservative form of Willmore surfaces equation 142
written in isothermal coordinates. A further corollary of theorem X.7 giving the Willmore surfaces equations in isothermal coordinates is the following. Corollary X.4. A conformal immersion Φ ⃗ of the flat disc D 2 is Willmore if and only if [ div ∇H ⃗ −3π ⃗n (∇H)+⋆(∇ ⃗ ⊥ ⃗n∧ H) ⃗ ] = 0 (X.86) where the operators div, ∇ and ∇ ⊥ are taken with respect to the flat metric 53 in D 2 . ✷ In order to exploit analytically equation (X.86) we will need a more explicit expression of π ⃗n (∇ ⃗ H). Let be the interior multiplication between p− and q−vectors p ≥ q producing p− q−vectors in R m such that (see [Fe] 1.5.1 combined with 1.7.5) : for every choice of p−, q− and p−q−vectors, respectively α, β and γ the following holds 〈α β,γ〉 = 〈α,β ∧γ〉 . Let (⃗e 1 ,⃗e 2 ) be an orthonormal basis of the orthogonal 2-plane to the m−2 plane given by ⃗n and positively oriented in such a way that ⋆(⃗e 1 ∧⃗e 2 ) = ⃗n and let (⃗n 1···⃗n m−2 ) be a positively oriented orthonormal basis of the m−2-plane given by ⃗n satisfying ⃗n = ∧ α ⃗n α . One verifies easily that ⎧ ⃗n ⃗e i = 0 ⎪⎨ ⎪⎩ ⃗n ⃗n α = (−1) α−1 ∧ β≠α ⃗n β ⃗n (∧ β≠α ⃗n β ) = (−1) m+α−2 ⃗n α 53 divX = ∂ x1 X 1 +∂ x2 X 2 , ∇f = (∂ x1 f,∂ x2 f) and ∇ ⊥ f = (−∂ x2 f,∂ x1 f). 143
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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
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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
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 and 122: X.4.2 Li-Yau Energy lower bounds an
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: the Willmore Lagrangian ”controls
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
Page 173 and 174: isabilipschitzdiffeomorphismbetween
Page 175 and 176: Having defined weak Willmore immers
Page 177 and 178: Let µ be the solution of ⎧ ⎨
Page 179 and 180: Combining this inequality with (X.1
Page 181 and 182: ii) iii) iv) limsupArea( Φ ⃗ k (
Page 183 and 184: The assumption i) implies that, mod
Page 185 and 186: Theorem X.14. Let ⃗ Φ be a Lipsc
Page 187 and 188: Thus < ∇ ⃗ Φ,∇ ⊥ ⃗ L >=
Page 189 and 190: To that purpose we first compute
Page 191 and 192: 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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