Conformally Invariant Variational Problems. - SAM

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where [p 1 ,p 2 ] is the segment joining the two points p 1 and p 2 in D 2 and dσ is the length element along this segment. The mean-value theorem implies then that there exists a point p 0 ∈ [p 1 ,p 2 ] such that λ(p 0 ) ≥ log |⃗ Φ(p 1 )− ⃗ Φ(p 2 )| |p 2 −p 1 | . Thus in particular we have ν(p 0 ) = (λ−µ)(p 0 ) ≥ log |⃗ Φ(p 1 )−Φ(p ⃗ 2 )| |p 2 −p 1 | ∫ − C |∇⃗n ⃗Φ | 2 . D 2 (X.172) Let ρ < 1 such that B ρ (0) contains the segment [p 1 ,p 2 ]. Let r := (1+ρ)/2. The explicit expression of the Poisson Kernel 67 gives ν(p 0 ) = r2 −|p 0 | 2 2πr ∫ ∂B 2 r (0) ν(z) |z −p 0 | dσ(z) , where dσ(z) is the volume form on ∂Br 2(0). Let ν+ = sup{0,ν} and ν − = inf{0,ν}. We then have ∫ ν − (z) 2πr dσ(z) ≥ ∂Br 2(0) |z −p 0 | r 2 −|p 0 | ν(p 0) 2 ∫ ν + (X.173) (z) − |z −p 0 | dσ(z) ∂B 2 r(0) We apply (X.165) on Br 2 (0) and we have ν + ≤ 1 [ ∫ ] 2 log+ C r e 2λ +‖µ‖ ∞ . D 2 67 See for instance [GT] theorem 2.6. (X.174) 178
Combining this inequality with (X.170) and (X.173) gives ∫ ν − (z) ∂Br 2(0) |z −p 0 | dσ(z) ≥ −C r ∣ log |⃗ Φ(p 1 )−Φ(p ⃗ 2 )| |p 2 −p 1 | ∣ [ ∫ ] (X.175) −C r log + C r e 2λ −C r |∇⃗n ⃗Φ | D ∫D 2 . 2 2 We deduce fromthisinequalityandfrom(X.174)combinedwith (X.170) that ∫ ∣ ∣∣∣∣ |ν| dσ ≤ C r log |⃗ Φ(p 1 )− ⃗ Φ(p 2 )| ∂B |p r(0) 2 2 −p 1 | ∣ [ ∫ ] (X.176) +C r log + C r e 2λ +C r |∇⃗n ⃗Φ | D ∫D 2 . 2 2 Using again the explicit expression of the Poisson Kernel, we have that for any p in B 2 ρ (0) ν(p) = r2 −|p| 2 2πr ∫ ∂B 2 r(0) ν(z) |z −p| dσ(z) , For any point p in B 2 ρ (0) and any point z in ∂B2 r (0), (r 2 −|p| 2 )/2πr|z −p| is bounded from above and from below by constants which only depend on ρ. Thus there exists C ρ > 0 such that ∣ ∣∣∣∣ ‖ν‖ L∞ (Bρ 2(0)) ≤ C ρ log |⃗ Φ(p 1 )−Φ(p ⃗ 2 )| |p 2 −p 1 | ∣ [ ∫ ] ∫ (X.177) +C ρ log + C r e 2λ +C ρ |∇⃗n ⃗Φ | 2 D 2 D 2 The combination of (X.170) and (X.177) gives the inequality (X.166) and theorem X.13 is proved. ✷ 179
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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
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Thus ⃗G ∗ ω S m−1(p) = 1 |S
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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 and 174: isabilipschitzdiffeomorphismbetween
Page 175 and 176: Having defined weak Willmore immers
Page 177: Let µ be the solution of ⎧ ⎨
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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Conformally Invariant Variational Problems. - SAM

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