Conformally Invariant Variational Problems. - SAM

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Assume that at p the vector H ⃗ ⃗Φ is parallel to ⃗n 1 , one has, using the coordinates s we introduced, ∫ ∣ ∣< G, ⃗ H ⃗ ∫ ⃗Φ > ∣ 2 Ω = s 2 1 |⃗ H ⃗Φ | 2 ω S m−3 S m−3 Fiber = |⃗ H ⃗Φ | 2 m−2 m−2 ∑ j=1 ∫S m−3 s 2 j ω S m−3 = |⃗ H ⃗Φ | 2 m−2 Denote N + Σ 2 the subset of NΣ 2 on which det non-negative. We have then proved ∫ G ⃗ ∗ ω S m−1 ≤ 1 ∫ N + Σ 2π 2 . ( < G, ⃗ ⃗ ) I > Σ 2 | ⃗ H ⃗Φ | 2 dvol ⃗Φ∗ g R m . (X.57) Now we claim that each points in S m−1 admits at least two preimages by G ⃗ ( at which det < G, ⃗ ⃗ ) I > ≥ 0 unless the immersed surface is included in an hyperplane of R m . Indeed, let ξ ⃗ ∈ S m−1 and consider the affine hyper-plane Ξ a given by < x, ξ ⃗ >= a Let and a + = max{a ∈ R ; Ξ a ∩ ⃗ Φ(Σ 2 ) ≠ ∅} a − = min{a ∈ R ; Ξ a ∩ ⃗ Φ(Σ 2 ) ≠ ∅} Assume a + = a − then the surfaced is immersed in the hyperplane Ξ a+ = Ξ a− and thus the claim is proved. If a + > a − , it is clear that ⃗ Φ(Σ 2 ) is tangent to Ξ a+ at a point ⃗ Φ(p + ) and hence ⃗ξ belongs to the normal space to ⃗ Φ ∗ T p Σ 2 which means in other words that ξ ⃗ ∈ G(N ⃗ p+ Σ 2 ). Similarly, Φ(Σ ⃗ 2 ) is tangent to Ξ a− at a point Φ(p ⃗ − ) and hence ξ ⃗ ∈ G(N ⃗ p− Σ 2 ). Since a − < a + , ⃗Φ(p − ) ≠ Φ(p ⃗ + ) and then we have proved that ξ ⃗ admits at least two prei-mages by G. ⃗ ( We claim now that det < G, ⃗ ⃗ ) I > ≥ 0 at these points. Since the whole surface is contained in one of 124 is
the half space given by the affine plane ( Φ(p ⃗ ± ), ξ) ⃗ we have that p →< ξ, ⃗ Φ(p)− ⃗ Φ(p ⃗ ± ) > has an absolute maximum (for +) or minimum (for −) at p = ±p. In both cases the determinant of the 2 by 2 Hessian has to be non-negative : det (< ξ,∂ ⃗ ) x 2 ⃗ i ,x jΦ > ≥ 0 from which we deduce, since ξ ⃗ is normal to Φ ∗ T p± Σ 2 , ( det < ξ, ⃗ ⃗ ) I > ≥ 0 Assume R m is the smallest affine subspace in which Φ(Σ ⃗ 2 ) is included. Then , using Federer’s co-area formula, the claim implies ∫ ∫ 2 ≤ #{ G ⃗−1 ( ξ), ⃗ ξ ∈ N + Σ 2 } ω S m−1( ξ) ⃗ = G ⃗ ∗ ω S m−1 . S m−1 N + Σ 2 (X.58) Combining (X.57) and (X.58) gives (X.52) : ∫ 4π ≤ | H| ⃗ 2 dvol ⃗Φ∗ g R Σ 2 m Assume this inequality is in fact an equality. Then all the inequalities above are equalities and in particular we have that tr g (< G, ⃗ ⃗ I >) 2 ( = det < G, ∣ 2 ∣ ⃗ ⃗ ) I > which implies the fact that the immersed surface is totally umbilic and it has then to be a translation of a sphere homothetic to S 2 the unit sphere of R 3 ⊂ R m . This concludes the proof of theorem X.2. ✷ The first part of theorem X.2 is in fact a special case of this more general result due to P. Li and S.T. Yau (see [LiYa]). 125
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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
Page 73 and 74: Just as in the proof of the regular
Page 75 and 76: VIII A proof of Heinz-Hildebrandt
Page 77 and 78: maps, namely ∑n+1 ( −∆u i =
Page 79 and 80: Theorem VIII.2. [Riv1] Let N n be a
Page 81 and 82: If A is almost everywhere invertibl
Page 83 and 84: the whole regularity result stated
Page 85 and 86: Bringing altogether (VIII.15), (VII
Page 87 and 88: In the simpler case when Ω is dive
Page 89 and 90: 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: Thus ⃗G ∗ ω S m−1(p) = 1 |S
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
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Having defined weak Willmore immers
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Let µ be the solution of ⎧ ⎨
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Combining this inequality with (X.1
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ii) iii) iv) limsupArea( Φ ⃗ k (
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The assumption i) implies that, mod
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Theorem X.14. Let ⃗ Φ be a Lipsc
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Thus < ∇ ⃗ Φ,∇ ⊥ ⃗ L >=
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To that purpose we first compute
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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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