Conformally Invariant Variational Problems. - SAM

More documents

Recommendations

Info

The definition of the Gauss curvature gives K ⃗n = − e−2λ 2 ∇⃗n×∇ ⊥ ⃗n = − e−2λ 2 Inserting (X.95) in (X.94) gives −4e 2λ ⃗ H (H 2 −K)+∇H ·[∇⃗n+⃗n×∇ ⊥ ⃗n ] This becomes = Hdiv [ ∇⃗n−⃗n×∇ ⊥ ⃗n ] . −4e 2λ H ⃗ (H 2 −K)−2∆H ⃗n [ = div −2∇H ⃗n+H ∇⃗n− H ⃗ ] ×∇ ⊥ ⃗n div [ ⃗n×∇ ⊥ ⃗n ] (X.95) . (X.96) (X.97) Usingnow”intrinsicnotations”(independentoftheparametrization) on (Σ 2 ,g), (X.97) says 4 H ⃗ (H 2 −K)+2∆ g H ⃗n [ = d ∗ g −2dH ⃗n+Hd⃗n−∗ H ⃗ ] ×d⃗n . (X.98) which is the desired identity (X.85) and theorem X.7 is proved in codimension 1. ✷ Before to proceed to the proof of theorem X.7 in arbitrary codimensionwe first introducesome complexnotationsthatwill be useful in the sequel. Assume Φ ⃗ is a conformal immersion into R m , one denotes z = x 1 + ix 2 , ∂ z = 2 −1 (∂ x1 − i∂ x2 ), ∂ z = 2 −1 (∂ x1 +i∂ x2 ). Moreover we denote 55 ⎧ ⎨ ⃗e z := e −λ ∂ zΦ ⃗ = 2 −1 (⃗e 1 −i⃗e 2 ) ⎩ ⃗e z := e −λ ∂ zΦ ⃗ = 2 −1 (⃗e 1 +i⃗e 2 ) 55 Observe that the notation has been chosen in such a way that ⃗e z =⃗e z . 148
Observe that ⎧⎪ ⎨ ⎪ ⎩ 〈⃗e z ,⃗e z 〉 = 0 〈⃗e z ,⃗e z 〉 = 1 2 (X.99) ⃗e z ∧⃗e z = i 2 ⃗e 1 ∧⃗e 2 Introduce moreover the Weingarten Operator expressed in our conformal coordinates (x 1 ,x 2 ) : ⃗H 0 := 1 [ ⃗I(e1 ,e 1 )− 2 ⃗ I(e 2 ,e 2 )−2i ⃗ I(e 1 ,e 2 )] . With these notations the following lemma holds Lemma X.3. Let Φ ⃗ be a conformal immersion of D 2 into R m π T (∂ zH)−i ⃗ ⋆(∂z ⃗n∧ H) ⃗ 〈〉 = −2 ⃗H, H0 ⃗ ∂ zΦ ⃗ (X.100) and hence ∂ zH ⃗ −3π⃗n (∂ zH)−i ⃗ ⋆(∂z ⃗n∧ H) ⃗ 〈〉 = −2 ⃗H, H0 ⃗ ∂ zΦ−2π⃗n ⃗ (∂ zH) ⃗ (X.101) Remark X.2. Observe that with these complex notations the identity (X.85), which reads in conformal coordinates [ − e−2λ 2 div ∇H ⃗ −3π ⃗n (∇H)+⋆(∇ ⃗ ⊥ ⃗n∧ H) ⃗ ] (X.102) = ∆ ⊥H ⃗ + Ã( H)−2| ⃗ H| ⃗ 2 H ⃗ , becomes 〈〉 4e −2λ R (∂ z [π ⃗n (∂ zH)+ ⃗ ⃗H, H0 ⃗ ∂ z ⃗ Φ ]) = ∆ ⊥ ⃗ H + Ã( ⃗ H)−2 | ⃗ H| 2 ⃗ H , (X.103) 149 ✷ ✷
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 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: where e λ = |∂ x1Φ| ⃗ = |∂x
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?