Conformally Invariant Variational Problems. - SAM

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CombiningthislemmawiththeequicontinuityprovedinlemmaV.2 we obtain that the weak limit of a minimizing sequence satisfies all the conditions for the membership in C ∗ (Γ) and then realizes a minimizer of (V.3). LemmaV.4isaconsequence ofpropositionV.1, corollaryV.1 and the following lemma that will be useful in the sequel. Lemma V.5. Let u be an harmonic function in W 1,2 (D 2 ,R m )∩ C 0 (∂D 2 ,R m ), then u ∈ C 0 (D 2 ,R m ). ✷ Proof of lemma V.3. Since u is harmonic into R m , i.e. each coordinate function u i satisfies ∆u i = 0, then u is smooth in the interior of the disc D 2 . It remains to prove the continuity at an arbitraryboundary point p ∈ ∂D 2 . Let ε > 0 and consider δ > 0 to be fixed later on in the proof. Applying Courant Lemma V.3 we obtain the existence of ρ ∈ [δ, √ δ] such that ∫ ∂B ρ (p)∩D 2 |∇u| dθ ≤ √ 4π log1/δ ∫ D 2 |∇u| 2 dx . The circle ∂B ρ (p) intersects ∂D 2 that we denote p(ρ) ± indexed in such a way that the positively oriented arc p+ (ρ)p ̂ − (ρ) in ∂D 2 contains the point p. The map u is in W 1,1 (∂B ρ (p)∩D 2 ) it admits limits u + and u − respectively at the points p + (ρ) and p − (ρ). One point has not been addressed at this stage in order to prove theorem V.2, this is the question to know whether C(Γ) is empty or not while assuming Γ to be a rectifiable closed curve only. We are going to answer positively to this question. 40
V.3 Existence of Parametric disc extensions to Jordan rectifiable curves. One point has not been addressed at this stage in order to prove theoremV.2, thisis the question to knowwhether C(Γ) is empty or not while assumingΓto be a rectifiableclosed curve only. We are going to answer positively to this question. Theorem V.3. Let Γ be a closed Jordan rectifiable curve then C(Γ) is not empty. ✷ Our approach consists in approximating Γ by smooth curves andpasstothelimitbutbeforetodosoweshallfirstestablishan isoperimetric inequality for smooth conformal parametrization. 41
Page 1 and 2: Conformally Invariant Variational P
Page 3 and 4: II Conformal transformations - some
Page 5 and 6: this implies ∀X,Y ∈ R m \{0} if
Page 7 and 8: This form is called the Hopf differ
Page 9 and 10: ∂ ∂x k (e 2λ δ ij ) = ∂ ∂
Page 11 and 12: Let k ∈ {1,...,n} and choose i
Page 13 and 14: If neither A = 0 nor B = 0 The left
Page 15 and 16: ✷ Proof of Lemma II.4 Since u(0)
Page 17 and 18: III Elementary Differential geometr
Page 19 and 20: control it gives on the image itsel
Page 21 and 22: This is the main advantage of worki
Page 23 and 24: oundary, and sending ∂D 2 monoton
Page 25 and 26: Such a Jordan curve is also simply
Page 27 and 28: Therefore, for m = 3, any minimizer
Page 29 and 30: critical point for variations in th
Page 31 and 32: Remark V.3. There are situations wh
Page 33 and 34: Thus the flow x t of the vector-fie
Page 35 and 36: Another difficulty lies in the rema
Page 37 and 38: and ‖u(x)−u(y)‖ 2 L ∞ ((∂
Page 39: in the middle of this arc : |p −
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 and 90: 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
Page 111 and 112:
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
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where we used that the unit sphere
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Conjecture X.1. Let ⃗ Φ be an im
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dle to Φ(Σ ⃗ 2 ) : for all X
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Let TR m ⃗ Φ([0,1]×Σ 2 ) be th
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imply D ∂ ⃗H = 1 ∂t 2 2∑ D
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We have moreover ∇ es ⃗ V N = =
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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 (
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 >=
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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. ”
Page 207 and 208:
[DHKW1] Dierkes, Ulrich; Hildebrand
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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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