Conformally Invariant Variational Problems. - SAM

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nalized in an orthonormal basis and the two eigenvalues κ 1 and κ 2 are called the principal curvatures of the surface at p. The mean curvature is then given by H := κ 1 +κ 2 2 and the mean curvature vector is given by ⃗H := H ⃗n = 1 2 tr(g−1 ⃗ I) = 1 2 2∑ g ij ⃗ I(∂xi ,∂ xj ) , ij=1 (X.3) where (x 1 ,x 2 ) are arbitrary local coordinates and (g ij ) ij is the inverse matrix to (g(∂ xi ,∂ xj )). In particular if (⃗e 1 ,⃗e 2 ) is an orthonormal basis of T p S, (X.3) becomes ⃗H = ⃗ I(⃗e 1 ,⃗e 2 )+ ⃗ I(⃗e 2 ,⃗e 2 ) . (X.4) 2 The Gauss curvature is given by ( ) det ⃗n·⃗I(∂ xi ,∂ xj ) K := = κ 1 κ 2 . (X.5) det(g ij ) The Willmore functional of the surface Σ is given by ∫ W(S) = | H| ⃗ 2 dvol g = 1 ∫ |κ 1 +κ 2 | 2 dvol g 4 S One can rewrite this energy in various ways and get various interpretations of this energy. Assume first that Σ is closed (compact without boundary) the Gauss-Bonnet theorem 33 asserts that the integralof K dvol g is proportionalto a topological invariant of S : χ(S) the Euler class of S. Precisely one has ∫ ∫ K dvol g = κ 1 κ 2 dvol g = 2π χ(S) S S (X.6) = 4π (1−g(S)) , 33 See for instance [doC1] S 104
where g(S) denotes the genus of S. Combining the definition of W and this last identity one obtains 34 W(S)−π χ(S) = 1 ∫ (κ 2 1 4 +κ2 2 ) dvol g S = 1 ∫ | 4 ⃗ I| 2 dvol g = 1 ∫ |d⃗n| 2 g dvol g . S 4 S (X.7) Hence modulo the addition of a topological term, the Willmore energy corresponds to the Sobolev homogeneous Ḣ1 −energy of the Gauss map for the induced metric g. Consider now, again for a closed surface, the following identity based again on Gauss-Bonnet theorem (X.6) : W(S) = 1 ∫ ∫ (κ 1 −κ 2 ) 2 dvol g + κ 1 κ 2 dvol g 4 S S = 1 ∫ (X.8) (κ 1 −κ 2 ) 2 dvol g +2π χ(S) . 4 S Hence, modulo this time the addition of the topological invariant of the surface 2πχ(S), the Willmore energy identifies with an energy that penalizes the lack of umbilicity when the two principal curvatures differ. The Willmore energy in this form is commonly called Umbilic Energy. Finally there is an interesting last expression of the Willmore energy that we would like to give here. Consider a conformal parametrization ⃗ Φ of the surface S for the conformal structure induced by the metric g (it means that we take a Riemann surface Σ 2 and a conformal diffeomorphism ⃗ Φ from Σ 2 into S). In 34 The last identity comes from the fact that, at a point p, taking an orthonormal basis (⃗e 1 ,⃗e 2 ) of T p S one has, since < d⃗n,⃗n >= 0 : |d⃗n| 2 g = 2∑ i,j=1 < d⃗n· ⃗e i ,⃗e j > 2 = 2∑ | ⃗ I(⃗e i ,⃗e j )| 2 = | ⃗ I| 2 . i,j=1 105
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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
Page 53 and 54: acting on maps u ∈ W 1,2 (D 2 ,N
Page 55 and 56: Whence, [ ∆u−H(u)(∇ ⊥ u,∇
Page 57 and 58: VII Integrabilitybycompensationtheo
Page 59 and 60: Accordingly, if φ lies in L ∞ ,
Page 61 and 62: Proof of the regularity of the solu
Page 63 and 64: 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: form associated to g on S at the po
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
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The tangential projection gives 4e
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X.6 Construction of Isothermal Coor
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the Lie algebra iR. Sections of thi
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R m is given by ∇ X σ := π T (d
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Let Σ 2 be a smoothcompactoriented
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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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Conformally Invariant Variational Problems. - SAM

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