Conformally Invariant Variational Problems. - SAM

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Lemma V.1. Let P 1 , P 2 and P 3 be 3 distinct points on ∂D 2 indexed in a trigonometric order then there is a unique element f ∈ M + (D 2 ), f(z) = e iθ (z −a)/(1−az) such that f(e 2ikπ/3 ) = P k for k = 1,2,3. ✷ We are now going to prove the closure of C ∗ (Γ) for the sequential weak W 1,2 topology. This closure implies the existence of a minimizer of E in C ∗ (Γ) and hence in C(Γ) too. Precisely we have the following theorem. Lemma V.2. For any positive constant C ≥ inf C∗ (Γ)E(u), the trace on ∂D 2 of the subset of elements u in C ∗ (Γ) satisfying E(u) ≤ C is equicontinuous. ✷ From an equicontinuous sequence one can always extract a subsequencethatuniformlyconverges(Arzelà-Ascoli’stheorem). Hence one deduces from this lemma the following corollary. Corollary V.1. Let u n be a sequence in C ∗ (Γ) weakly converging to a map u ∞ in W 1,2 then u ∞ is continuous and monotone on ∂D 2 . ✷ The main toolwe shall use in orderto prove the lemma V.2 is an argument introduced in the framework of the Plateau problem by R. Courant. The argument is based on Fubini theorem in order to extract a ”good slice”, an arc of circle on which the energy is controlled and on which we can apply Sobolev embedding which is better in this one dimensional curve than on the whole 2-dimensional disc. Lemma V.3. [Courant Lemma] Let u ∈ W 1,2 (D 2 ,R m ) and let p ∈ ∂D 2 . For any 0 < δ < 1. Then there exists ρ ∈ [δ, √ δ] such that ∇u ∈ L 2 (∂B ρ (p)∩D 2 ) , 36
and ‖u(x)−u(y)‖ 2 L ∞ ((∂B ρ (p)∩D 2 ) 2 ) ≤ [ ∫ ∂B ρ (p)∩D 2 |∇u| dθ ≤ 4π log 1 δ ∫ D 2 |∇u| 2 dx where dθ is the length form on ∂B ρ (p)∩D 2 . ] 2 (V.27) ✷ Proof of lemma V.3. Using Fubini theorem we have ∫ ∫ |∇u| 2 dx ≥ |∇u| 2 dx D 2 ≥ ∫ √ δ δ ∫ dρ { ∫ ≥ essinf ρ D 2 ∩(B √ δ (p)\B δ(p)) D 2 ∩∂B ρ (p) D 2 ∩∂B ρ (p) |∇u| 2 dθ |∇u| 2 dθ } ∫ √ δ Thus there exists a radius ρ ∈ [δ, √ δ] such that ∫ ρ |∇u| 2 dθ ≤ 2 ∫ D 2 ∩∂B ρ (p) log 1 |∇u| 2 dx δ D 2 δ dρ ρ (V.28) Cauchy-Schwartz inequality gives [ ∫ 2 ∫ |∇u| dθ] 2 ≤ |D 2 ∩ ∂B ρ (p)| |∇u| 2 dθ . D 2 ∩∂B ρ (p) D 2 ∩∂B ρ (p) (V.29) Since |D 2 ∩ ∂B ρ (p)| ≤ 2πρ, combining (V.28) and (V.29) gives (V.27) and lemma V.3 is proved. ✷ Proof of lemma V.2. Let C > 0 satisfying C ≥ inf C ∗ (Γ) E(u) . 37
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: Another difficulty lies in the rema
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
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In the simpler case when Ω is dive
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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
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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 (
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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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