Conformally Invariant Variational Problems. - SAM

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gauge invariant L 2 −norm of the curvature, provided this energy is below some universal threshold. Similarly in the present situation the Coulomb Gauge - or conformal choice of coordinates - will permit to control the L ∞ norm of the pull-back metric (the conformal factor) with the help of the ”gauge invariant” L 2 -norm of the second fundamental form provided it stays below the universal threshold : √ 8π/3. This will be the up-shot of the present subsection. This L ∞ control of the conformal factor will be an application of integrability of compensation and Wente estimates more specifically. X.6.1 The Chern Moving Frame Method. WepresentamethodoriginallyduetoS.S.Cherninordertoconstruct local isothermal coordinates. It is based on the following observation. Let ⃗ Φ be a conformal immersion of the disc D 2 introduce the following tangent frame : (⃗e 1 ,⃗e 2 ) = e −λ (∂ x1 ⃗ Φ,∂x2 ⃗ Φ) , where e λ = |∂ x1 ⃗ Φ| = |∂x2 ⃗ Φ|. A simple computation shows 〈⃗e 1 ,∇⃗e 2 〉 = −∇ ⊥ λ , (X.127) and in particular it follows div〈⃗e 1 ,∇⃗e 2 〉 = 0 . (X.128) This identity can be writen independently of the parametrization as follows d ∗ g 〈⃗e 1 ,d⃗e 2 〉 = 0 . (X.129) It appears clearly as the Coulomb condition : cancelation of the codifferential of the connection on the S 1 −tangent orthonormal frame bundle given by the 1-form i〈⃗e 1 ,d⃗e 2 〉 taking value into 158
the Lie algebra iR. Sections of this bundles are given by maps ( ⃗ f 1 , ⃗ f 2 ) from D 2 into ⃗ Φ ∗ (TD 2 × TD 2 ) such that ( ⃗ f 1 , ⃗ f 2 )(x 1 ,x 2 ) realizes a positive orthonormal basis of ⃗ Φ ∗ (T (x1 ,x 2 )D 2 ). Thepassagefromonesection(⃗e 1 ,⃗e 2 )toanothersection( ⃗ f 1 , ⃗ f 2 ) is realized through a change of gauge which corresponds to the actionofanSO(2)rotatione iθ onthetangentspace ⃗ Φ ∗ (T (x1 ,x 2 )D 2 ) : ⃗f 1 +i ⃗ f 2 = e iθ (⃗e 1 +i⃗e 2 ) . The expression of the same connection but in the new trivialization given by the section ( f ⃗ 1 , f ⃗ 2 ) satisfies the classical gauge change formula for an S 1 -bundle : 〈 i ⃗f1 ,df ⃗ 〉 2 = i〈⃗e 1 ,d⃗e 2 〉+idθ . (X.130) The curvature of this connection is given by id〈⃗e 1 ,d⃗e 2 〉 = i[< D e1 ⃗e 1 ,D e2 ⃗e 2 > − < D e2 ⃗e 1 ,D e1 ⃗e 2 >] e ∗ 1 ∧e∗ 2 [ = i < ⃗ I(⃗e 1 ,⃗e 1 ), ⃗ I(⃗e 2 ,⃗e 2 ) > −| ⃗ I(⃗e 1 ,⃗e 2 )| 2] (X.131) e ∗ 1 ∧e ∗ 2 = i K dvol g where we recall the notations we already introduced : e i is the vector field on D 2 given by d ⃗ Φ · e i = ⃗e i , D ei ⃗e j := π ⃗n (d⃗e j · e i ) and K is the Gauss curvature of (D 2 , ⃗ Φ ∗ g). In the last identity we have made use of Gauss theorem (theorem 2.5 chap. 6 in [doC2]). Combining (X.127) and (X.131) gives the well known expression of the Gauss curvature in isothermal coordinates in terms of the conformal factor λ : −∆λ =< ∇ ⊥ ⃗e 1 ,∇⃗e 2 >= e 2λ K . (X.132) 159
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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.
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 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: X.6 Construction of Isothermal Coor
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
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
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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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