Conformally Invariant Variational Problems. - SAM

More documents

Recommendations

Info

vectors are denoted ⃗e i (t,p). The fact that this transport is parallel w.r.t. ∇ implies that (⃗e 1 (t,p),⃗e 2 (t,p)) realizes a positive orthonormal basis 48 of ⃗ Φ ∗ (t×T p Σ 2 ). Next we extend ⃗e i (t,p) parallely and locally in {t}×Σ 2 with respect again to ∇ along geodesics in {t} × Σ 2 starting from (t,p) for the induced metric ⃗ Φ ∗ t g R m. We will denote also e i := ( ⃗ Φ −1 t ) ∗ ⃗e i . By definition one has ⃗H(t,p) := 1 2 2∑ π ⃗n (d⃗e s ·e s ) = 1 2 s=1 2∑ D es ⃗e s , wherewehaveextendedtheuseofthenotationDforanysection of T ⊕N in an obvious way 49 . Hence we have D ∂ ⃗H = 1 ∂t 2 2∑ s=1 s=1 D ∂ (D es ⃗e s ) (X.69) ∂t Since on ⃗ Φ([0,1] × {p}) one has ∇⃗e s ≡ 0 and since moreover ∇+D coincides with the differentiation 50 d in R m one has D ∂(D es ⃗e s )(0,p) = D ∂ (d⃗e s ·e s )(0,p) ∂t ∂t = π ⃗n (d(∂ t ⃗e s )·e s )+π ⃗n (d⃗e s ·∂ t e s ) (X.70) Observe that for a fixed q ∈ Σ 2 e s (t,q) stays in T q Σ 2 as t varies and then there isno need of a connectionto define ∂ t e s . Observe moreover that ∂ t e s = [∂ t ,e s ]. Thus (X.69) together with (X.70) 48 Indeed ( ∇ ∂ ⃗e i = 0 ⇒ π T d⃗e i · ∂ ) = 0 ∂t ∂t 〈〉 ∂⃗ei ⇒ ∀i,j ∂t ,⃗e j = 0 ⇒ ∀i,j ∂ ∂t = 0 . 49 D X σ := π ⃗n (dσ ·X). D does not realizes a connection on the whole T ⊕N. 50 d is the flat standard connection on TR m 134
imply D ∂ ⃗H = 1 ∂t 2 2∑ D es (∂ t ⃗e s )+ 1 2 s=1 We denote V := ∂ t ⃗ Φ. Observe that 2∑ D [∂t ,e s ]⃗e s . s=1 (X.71) ∂ t ⃗e s = ∂ t (d ⃗ Φ·e s ) = d ⃗ V ·e s +d ⃗ Φ t ·∂ t e s = d ⃗ V ·e s +d ⃗ Φ·[∂ t ,e s ] = d ⃗ V ·e s +[ ⃗ V,⃗e s ] . (X.72) Observe moreover that for two tangent vector-fields X and Y on Σ 2 one has 51 D X (dΦ·Y) ⃗ = D Y (dΦ·X) ⃗ . (X.73) This last identity implies in particular that D es ([ ⃗ V,⃗e s ]) = D [∂t ,e s ]⃗e s . (X.74) Combining (X.71), (X.72) and (X.74) gives We have D ∂ ⃗H = 1 ∂t 2 2∑ D es (dV ⃗ ·e s )+ s=1 2∑ D es [ V,⃗e ⃗ s ] . s=1 (X.75) [ ⃗ V,⃗e s ] = d ⃗ Φ·[∂ t ,e s ] = d⃗e s ·∂ t −d ⃗ V ·e s 51 Indeed in local coordinates (x 1 ,x 2 ) in Σ 2 one has D X (d ⃗ Φ·Y) = 2∑ k,j=1 π ⃗n (∂ xk (∂ xj ⃗ ΦYj )X k ) = ∑ 2 j,k=1 π ⃗n(∂ xj (∂ xk ⃗ ΦXk )Y j )− ∑ 2 j,k=1 π ⃗n(∂ xk ⃗ ΦYj ∂ xj X k ) = D Y (d ⃗ Φ·X) , where we have used the fact for all j,k = 1,2 ⃗n(∂ xk ⃗ ΦYj ∂ xj X k ) = 0 and ⃗n(∂ xj ⃗ ΦXk ∂ xk Y j ) . 135
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 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
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: Let TR m ⃗ Φ([0,1]×Σ 2 ) be th
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
Page 155 and 156: The tangential projection gives 4e
Page 157 and 158: X.6 Construction of Isothermal Coor
Page 159 and 160: the Lie algebra iR. Sections of thi
Page 161 and 162: R m is given by ∇ X σ := π T (d
Page 163 and 164: Let Σ 2 be a smoothcompactoriented
Page 165 and 166: There exists a constant C > 0, such
Page 167 and 168: such a way that ⃗n ρ λ realizes
Page 169 and 170: We make now use of (X.87) and we de
Page 171 and 172: We are now in positionto start the
Page 173 and 174: isabilipschitzdiffeomorphismbetween
Page 175 and 176: Having defined weak Willmore immers
Page 177 and 178: Let µ be the solution of ⎧ ⎨
Page 179 and 180: Combining this inequality with (X.1
Page 181 and 182: 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
Page 191 and 192:
and the following equation holds
Page 193 and 194:
One verifies easily that π T ( ⃗
Page 195 and 196:
From (X.197) we compute ⋆(⃗n
Page 197 and 198:
For any such a vector field X satis
Page 199 and 200:
X.7.4 The conformal Willmore surfac
Page 201 and 202:
Denote ∂ z ⃗ L = A⃗ez +B⃗e
Page 203 and 204:
We have using (X.113) ∂ z ∂ z
Page 205 and 206:
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
show all

Conformally Invariant Variational Problems. - SAM

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?