Conformally Invariant Variational Problems. - SAM

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Varitaions in the domain correspond to perturbation of the form x(s + tX) ≃ x(s) + tẋ(s) + o(t 2 ). Multiplying the Euler Lagrange equation by the infinitesimal perturbation ẋ(s) corresponding then to the variation in the domain gives the stationary equation 0 = ẋ(s) [mẍ(s)+V ′ (x(s))] = d [ m (s)+V(x(s))] ds 2 ẋ2 which is nothing but the conservation of energy. ✷ We shall now exploit the specificity of the boundary condition imposed by the membership of the minimizer u of E in C(Γ), whose existence is still assumed at this stage of the proof in order to get more information on the Hopf differential and the fact that H(u) is identically zero. This is the result of the free Dirichlet condition we are imposing. By imposing a fixed Dirichletconditiontherewouldhavebeennoreasonfortheholomorphic Hopf differential to be identically zero and hence the minimizer u to be conformal. Precisely we are now going to prove the following proposition. Proposition V.3. Let u be a map in W 1,2 (D 2 ,R m ) satisfying then ∀X ∈ C ∞ (D 2 ,R 2 ) s. t. X ·x ≡ 0 on ∂D 2 d dt E(u(id+tX)) | t=0 = 0 . (V.16) |∂ x1 u| 2 −|∂ x2 u| 2 −2i < ∂ x1 u,∂ x2 u >≡ 0 on D 2 . Proof of proposition V.3. Let X be an arbitrary smooth vector-field on the disc D 2 satisfying X ·x ≡ 0 on ∂D 2 . (V.17) 32 ✷
Thus the flow x t of the vector-field X preserves D 2 and (V.12) still holds. Thus we have also (V.13). The stationarity assumption (V.16) implies then [ ∫ 2∑ lim −|∇u| 2 divX +2 dx = 0 . r→1− Br(0) 2 i,j=1 (V.18) We adopt for X the complex notation X := X 1 + iX 2 and we observe that (V.18) becomes ∫ [ lim H(u) ∂X ] dx 1 ∧dx 2 = 0 , (V.19) r→1− ∂z which also reads B 2 r (0) R lim r→1− R ( i 2 or equivalently ∫ − lim r→1− R ( i 2 ∂ xi u∂ xj u ∂ xi X j ] B 2 r (0) H(u) ∂X ∂z dz ∧dz ) ∫ B 2 r(0) = 0 (V.20) ) H(u) dX ∧dz = 0 (V.21) From proposition V.2 we know that H(u) is holomorphic and then smooth in the interior of D 2 . We can then integrate by part and we obtain lim I X dH(u)∧dz − r→1− (∫Br(0) 2 ∫ ∂B 2 r(0) ) X H(u) dz = 0 . (V.22) But dH(u)∧dz = ∂ z H(u) dz ∧dz ≡ 0, thus we finally obtain ) lim I X H(u)(z) dz = 0 . (V.23) r→1− (∫∂Br(0) 2 We choose the vector field X to be of the form X = iαz where α(e iθ ) is an arbitrary real function independent of |z| in the 33
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: Remark V.3. There are situations wh
Page 35 and 36: Another difficulty lies in the rema
Page 37 and 38: 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 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
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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.
Page 107 and 108:
- 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
Page 113 and 114:
and that we have denoted by ( ⃗
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and similarly the second fundamenta
Page 117 and 118:
Let (⃗n 1 ,··· ,⃗n m−2 ) b
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orthonormal basis for the metric g.
Page 121 and 122:
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
Page 129 and 130:
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
Page 149 and 150:
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
Page 163 and 164:
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 ⎧ ⎨
Page 179 and 180:
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 >=
Page 189 and 190:
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
Page 203 and 204:
We have using (X.113) ∂ z ∂ z
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References [Ad] Adams, David R. ”
Page 207 and 208:
[DHKW1] Dierkes, Ulrich; Hildebrand
Page 209 and 210:
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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