Conformally Invariant Variational Problems. - SAM

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Combining (X.25) and (X.26) we obtain ∇ h Y X −∇g YX = dµ·X Y +dµ·Y X −g(X,Y) U , (X.27) where U ∈ T p Σ n is the dual vector to the 1-form dµ for the metric g which is given by Similarly we have ∇ h ⃗ Y ⃗ X − ⃗ ∇ g ⃗Y ∀Z ∈ T p Σ n g(U,Z) = dµ·Z . (X.28) ⃗X = dµ· ⃗X ⃗ Y +dµ· ⃗Y ⃗ X −g( ⃗ X, ⃗ Y) ⃗ U , (X.29) where ⃗ V ∈ T ⃗Φ(p) M m is the dual vector to the 1-form dµ for the metric g which is given by ∀ ⃗ Z ∈ T ⃗Φ(p) M m g( ⃗ V, ⃗ Z) = dµ· ⃗Z . (X.30) Combining (X.22), (X.23), (X.28) and (X.30) we obtain ⃗ I h (X,Y)− ⃗ I g (X,Y) = −g(X,Y) ⃗ W . (X.31) where ⃗ W := ⃗ V − ⃗ U. From (X.28) and (X.30) we obtain that ∀ ⃗ Z ∈ ⃗ Φ ∗ (T p Σ n ) g( ⃗ Z, ⃗ W) = 0 . hence ⃗ U is the orthogonal projection of ⃗ V onto ⃗ Φ ∗ (T p Σ n ) and ⃗W is the orthogonal projection onto the normal space to ⃗ Φ(Σ n ). Let ξ ⃗ be a unit normal vector to Σ n in (M m ,g). For any such a vector ξ, ⃗ the g−scalar product between ξ ⃗ and ⃗ I g at a point p ∈ Σ n produces a symmetric bilinear two-form g( ξ, ⃗ ⃗ I g (·,·)) on (T p Σ n ,g). We denote by κ g 1 (ξ) and κg 2 (ξ) the eigenvalues of this symmetric two form for the g scalar product. Let (e 1 ,··· ,e n ) be an orthonormal basis to (T p Σ n ,g), we have g ( ⃗ξ, ) n∑ ⃗ I g (e i ,e i ) i=1 = tr g (g( ξ, ⃗ ⃗ ) I g (·,·)) = κ 1 ( ⃗ ξ)+···+κ n ( ⃗ ξ) . 116 (X.32)
Let (⃗n 1 ,··· ,⃗n m−2 ) be an orthonormal basis for the metric g of the normal space to ⃗ Φ ∗ (T p Σ n ). Combining the expression of the mean curvature vector (X.20) together with (X.32) we obtain the following expression of the mean curvature vector ⃗ H g of the immersion ⃗ Φ into (M m ,g) : Hence ⃗H g = 1 n m−2 ∑ α=1 n∑ κ g i (⃗n α) ⃗n α . i=1 | H ⃗ g | 2 g = 1 m−2 ∑ n∑ 2 n 2 κ g i ∣ (⃗n α) ∣ α=1 i=1 = 1 ( m−2 ∑∑ κ g i (⃗n α)−κ g j (⃗n α) ) 2 n 2 n−1 α=1 i
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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
Page 65 and 66: where u still denotes the normal un
Page 67 and 68: frames, thereby compensating for th
Page 69 and 70: tangent frame field to T 2 . Define
Page 71 and 72: 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 similarly the second fundamenta
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 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
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such a way that ⃗n ρ λ realizes
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We make now use of (X.87) and we de
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We are now in positionto start the
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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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