Conformally Invariant Variational Problems. - SAM

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Using Codazzi-Mainardi identity (X.118) and using also again identity (X.113), (X.125) becomes ]) 4e −2λ R (∂ z [π ⃗n (∂ zH)+ ⃗ < H, ⃗ H0 ⃗ > ∂ zΦ ⃗ (〈〉 ) (X.125) = ∆ ⊥H ⃗ +2R ⃗H, H0 ⃗ ⃗H0 . The definition (X.65) of Ã gives 2∑ Ã( H) ⃗ = < H, ⃗ ⃗ h ij > ⃗ h ij i,j=1 hence a short elementary computation gives Ã( ⃗ H)−2| ⃗ H| 2 ⃗ H = 2 −1 〈 ⃗H, ⃗ h11 − ⃗ h 22 〉 ( ⃗ h 11 − ⃗ h 22 )+2 < ⃗ H, ⃗ h 12 > ⃗ h 12 Using ⃗ H 0 this expression becomes Ã( ⃗ H)−2| ⃗ H| 2 ⃗ H = 2R (〈 ⃗H, ⃗ H0 〉 ⃗H0 ) (X.126) Combining (X.125) and (X.126) gives (X.103) which is the desired inequality and theorem X.7 is proved. ✷ 156
X.6 Construction of Isothermal Coordinates. In the previous subsection we discussed the difficulty to work with the Willmore surfaces equation due to the huge invariance group given by the space of positive diffeomorphisms of the surface. A classical way to by-pass this difficulty consists in ”breaking” the symmetry group (or ”gauge group”) of coordinates by restricting to a special subclass satisfying the Coulomb condition. In the present subsection we will explain why this choice correspondstotheconformalcondition. Wehaveseenintheprevious subsection that, in such coordinates, the Willmore surface equations can be written in a conservative-elliptic form which is critical with respect to the L 2 −norm of the second fundamental form. This triggers the hope to give answers to the analysis questions we raised for Willmore surfaces. Breakingthesymmetrygroupofcoordinatesbytakingisothermal ones is however not enough per se. The uniformization theorem tells us that, taking the conformal class defined by the pull-back metric Φ ⃗ ∗ g R m on Σ 2 , there is a system of coordinates on the fundamental domain of either C∪∞, C or the Poincaré half-plane corresponding to this class in which our immersion is conformal. However, in a minimization procedure for instance, taking a minimizing sequence of immersions Φ ⃗ k , assuming the conformal class defined by Φ ⃗ k on Σ 2 is controlled - is not convergingtotheboundaryofthemodulispace-thereisa-priorino control of the conformal factor corresponding to the pull-back metric Φ ⃗ ∗ k g Rm and the fact that we are in conformal coordinates is not helping much. We need then to have Conformal coordinates + estimates of the conformal factor. InGaugetheorysuchasinYang-Millsproblemin4dimension for instance, the Coulomb choice of gauge is the one that provides estimates of the connection that will be controlled by the 157
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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
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 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: The tangential projection gives 4e
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
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. ”
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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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