Conformally Invariant Variational Problems. - SAM

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where Â and ˆB are arbitrary functions in L ∞ ∩ W 1,2 and in W 1,2 respectively. An analogousversion 27 of theorem VII.1 with NeumanboundaryconditionsinplaceofDirichletconditions,we deduce that the unique solution (Ã,B) of (VIII.36) satisfies the estimates ∫ ∫ |∇Ã|2 +‖Ã−Id m‖ 2 ∞ ≤ C D ∫D |∇Â|2 |∇ξ| 2 2 2 D ∫ 2 +C |∇ˆB| ∫D 2 |∇P| 2 , 2 D 2 (VIII.38) and ∫ ∫ |∇(˜B −B 0 )| 2 ≤ C ‖Â−Id m‖ 2 ∞ |∇ξ| 2 D 2 D ∫ 2 +C ∫D |∇Â|2 |∇P| 2 , 2 D 2 where B 0 is the solution in W 1,2 of ⎧ ⎨ ∆B 0 = −div(∇ξ P −1 ) in D 2 ⎩ Hence, if (VIII.39) B 0 = 0 on ∂D 2 (VIII.40) ∫ D 2 |∇P| 2 +|∇ξ| 2 issufficientlysmall(thiscanalwaysbearrangedowingto(VIII.35) and the hypothesis (VIII.25)), then a standard fixed point argument in the space ( L ∞ ∩W 1,2 (D 2 ,M m (R)) ) ×W 1,2 (D 2 ,M m (R)) 27 whose proof is left as an exercise. 90
yields the existence of the solution (Ã,B) of the system ⎧ ∆Ã = ∇Ã·∇⊥ ξ +∇ ⊥ B ·∇P ⎪⎨ ∆B = −div(Ã∇ξ P−1 )+∇ ⊥ Ã·∇P −1 ∂Ã = 0 and B = 0 on ∂D2 ∂ν ∫ ⎪⎩ Ã = π 2 Id m D 2 (VIII.41) By construction, this solution satisfies the estimate (VIII.26) with A = ÃP−1 . The proof of theorem VIII.4 will then be finished once it is established that (A,B) is a solution of (VIII.27). To do so, we introduce the following linear Hodge decomposition in L 2 : ∇Ã−Ã∇⊥ ξ +∇ ⊥ B P = ∇C +∇ ⊥ D where C = 0 on ∂D 2 . The first equation in (VIII.41) states that ∆C = 0, so that C ≡ 0 sur D 2 . The second equation in (VIII.41) along with the boundary conditions imply that D satisfies ⎧⎨ ⎩ div(∇D P −1 ) = 0 in ∂D 2 D = 0 on ∂D 2 . (VIII.42) Thus, there exists E ∈ W 1,2 (D 2 ,M n (R)) such that ⎧ ⎪⎨ −∆E = ∇ ⊥ D·∇P −1 in D 2 (VIII.43) ⎪⎩ ∂E ∂ν = 0 on ∂D2 91
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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 ∞ ((∂
Page 39 and 40: in the middle of this arc : |p −
Page 41 and 42: V.3 Existence of Parametric disc ex
Page 43 and 44: We further assume that L is conform
Page 45 and 46: We note that Γ i (∇u,∇u) :=
Page 47 and 48: is explicitly given by u(x,y) := lo
Page 49 and 50: Example 3. We consider a map (ω ij
Page 51 and 52: 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: Theorem VIII.5. [Uhl], [Riv1] Let m
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 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
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the Willmore Lagrangian ”controls
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written in isothermal coordinates.
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quantity div [ 2∇H ⃗ −3H∇
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where e λ = |∂ x1Φ| ⃗ = |∂x
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Observe that ⎧⎪ ⎨ ⎪ ⎩ 〈
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and consequently ⋆(⃗n∧∇ ⊥
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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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