Conformally Invariant Variational Problems. - SAM

More documents

Recommendations

Info

Lemma II.2. Let A ∈ L(R m ,R n ) the space of linear maps from R m into R n Assume that A ≠ 0. Then ∀X,Y ∈ R m < X,Y > R m= 0 (II.3) =⇒ < AX,AY > R n= 0 Where R k denotes the canonical scalar product on the euclidian space R k if and only if there exists λ ∈ R such that ∀X,Y ∈ R m < AX,AY > R n= e 2λ < X,Y > R m (II.4) Proof of Lemma II.2 We assume II.2 Let X ∈ R m , X ≠ 0 We introduce L x R m −→ R n given by ∀y ∈ R m | AX |2 L X Y : = < AX,AY > R n − < X,Y > | X | 2 R m Denote (X) ⊥ the subspace of R m made of Vectors Y orthgonal to X. We have ⎧⎪ ⎨ ⎪ ⎩ L X X = 0 L X (X) ⊥ = 0 RX ⊕(X) ⊥ = R m This clearly implies that L X ≡ 0 For any X and Y in R m \{0} we then have ⎧ ⎪⎨ ⎪⎩ < AX,AY > R n= < AX,AY > R n= |AY|2 |Y| 2 | AX |2 | X | 2 < X,Y > R m < X,Y > R m (II.4b) 4
this implies ∀X,Y ∈ R m \{0} if < X,Y >≠ 0 then we have | AX | 2 | X | 2 < X,Y > R m= | AX | | X | = | AY | | Y | if < X,Y >= 0 and X ≠ 0 and Y ≠ 0 Let Z = X+Y 2 < X,Z >≠ 0 and < Y,Z >≠ 0 and from (II.5) we deduce | AY |2 | Y | 2 < X,Y > R m (II.5) (II.6) | AX | | X | = | AZ | | Z | = | AY | | Y | Hence (II.6) holds for any pair of vectors in R m \{0} and since |AX| A ≠ 0 |X| = e λ for some λ ∈ R This implies (II.4) . It is clear that (II.4) implies (II.3) and we have proved then Lemma II.2 . ✷ Famousconformaltransformationsaregivenbytheholomorphic or the antiholomorphic maps from a domain C C into C. This characterizes uniquely the conformal transformations from a 2- dimentional domain of C into C. We have indeed the following result. Proposition II.1. Let u be a C 1 map from a connected domain U of C into C. U is conformal if and only if on U one has 5
Page 1 and 2: Conformally Invariant Variational P
Page 3: II Conformal transformations - some
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 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
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?