Conformally Invariant Variational Problems. - SAM

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where ⃗ I and ⃗n denote respectively the second fundamental form andtheGaussmapoftheconformalimmersionΦ◦Ψ. ⃗ Themean curvature vector of Φ◦Ψ, ⃗ that we simply denote H, ⃗ is given by ⃗H [ := 2 −1 e −2λ ⃗ I(∂x1 ,∂ x1 )+ ⃗ ] I(∂ x2 ,∂ x2 ) ∈ L 2 (D 2 ) . Hence we have that moreover, we have also ∇ ⃗ H ∈ H −1 (D 2 ) , ⋆(∇ ⊥ ⃗n∧ ⃗ H) ∈ L 1 (D 2 ) . Using the expression (X.88), we also deduce that π ⃗n (∇ ⃗ H) ∈ H −1 +L 1 (D 2 ) . Hence for any immersion ⃗ Φ in E Σ the quantity ∇ ⃗ H −3π ⃗n (∇ ⃗ H)+⋆(∇ ⊥ ⃗n∧ ⃗ H) ∈ H −1 +L 1 (D 2 ) defines a distribution in D ′ (D 2 ). Using corollary X.4 one can then generalize the notion of Willmore immersion that we defined for smooth immersions to immersions in E Σ in the following way. Definition X.4. Let Σbe asmoothcompactoriented2-dimensional manifold and let Φ ⃗ be a Lipschitz immersion with L 2 −bounded second fundamental form. ⃗Φ is called a weak Willmore immersion if, in any lipschitz conformal chart Ψ from D 2 into (Σ, Φ ⃗ ∗ g R m), the following holds [ div ∇H ⃗ −3π ⃗n (∇H)+⋆(∇ ⃗ ⊥ ⃗n∧ H) ⃗ ] = 0 in D ′ (D 2 ) . where ⃗ H and ⃗n denote respectively the mean curvature vector and the Gauss map of ⃗ Φ in the chart Ψ and the operators div, ∇ and ∇ ⊥ are taken with respect to the flat metric 65 in D 2 . ✷ 65 divX = ∂ x1 X 1 +∂ x2 X 2 , ∇f = (∂ x1 f,∂ x2 f) and ∇ ⊥ f = (−∂ x2 f,∂ x1 f). 174
Having defined weak Willmore immersion that will naturally come in our minimization procedure it is a fair question to ask whether solutions to this elliptic non-linear system for which we have seen that it is critical in two dimensions for the Willmore energyaresmooth. Orinotherwordsweareaskingthefollowing question : Are weak Willmore immersions smooth Willmore immersions ? We will answer positively to that question in the next sections. X.6.6 Isothermal Coordinates with Estimates. In the previous subsections we explained how to produce locally and globallyconformalparametrizations,Coulombgauges, for Lipschitz immersions with L 2 −bounded second fundamental forms. This has been obtained in a qualitative way, without any care of establishing estimates. The goal of the present subsection is to remedy to it in giving L ∞ controls of the metric in conformal parametrization by the mean of several quantities (L 2 −norm of the second fundamental form, area of the image, distance of the images of 2 distinct points) and under the assumption that the L 2 −norm of the second fundamental form is below some threshold. Precisely we shall prove the following result. Theorem X.13. [Control of local isothermal coordinates] Let Φ ⃗ be a lipschitz conformal immersion 66 from the disc D 2 into R m . Assume ∫ |∇⃗n ⃗Φ | 2 < 8π/3 . (X.164) D 2 66 A Lipschitz conformal map from D 2 into R m satisfying (X.144) 175
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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
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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
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
Page 171 and 172: We are now in positionto start the
Page 173: isabilipschitzdiffeomorphismbetween
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
Page 193 and 194: One verifies easily that π T ( ⃗
Page 195 and 196: From (X.197) we compute ⋆(⃗n
Page 197 and 198: For any such a vector field X satis
Page 199 and 200: X.7.4 The conformal Willmore surfac
Page 201 and 202: Denote ∂ z ⃗ L = A⃗ez +B⃗e
Page 203 and 204: We have using (X.113) ∂ z ∂ z
Page 205 and 206: References [Ad] Adams, David R. ”
Page 207 and 208: [DHKW1] Dierkes, Ulrich; Hildebrand
Page 209 and 210: larityof weak solutionsof nonlinear
Page 211 and 212: [Poi] Poisson, Siméon Denis ”Ext
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