Conformally Invariant Variational Problems. - SAM

More documents

Recommendations

Info

[LaRi] Lamm, Tobias; Rivière Tristan ”Conservation laws for fourth order systems in four dimensions”. Comm.P.D.E., 33 (2008), no. 2, 245-262. [LaLi] Landau, L. D.; Lifschitz, E. M. ”Theory of elasticity” - Course of theoretical physics, volume 7 - third edition revised and enlarged by E.M.Lifshitz, A.M.Kosevich and L.P.Pitaevskii, Butterwoth-Heinemann 1986. [LiYa] Li,P.,Yau,S.-T.,ANewConformal<strong>Invariant</strong>anditsApplications to the Willmore Conjecture and the First Eigenvalue on Compact Surfaces, Inventiones Math. 69 (1982), 269-291. [MaNe] Marques, F.C.; Neves, A. ”Min-Max theory and the Willmore conjecture”. arXiv:1202.6036 (2012). [Mor] Morgan, Frank ”Geometric measure theory. A beginner’s guide”. Fourth edition. Elsevier/Academic Press, Amsterdam, 2009. [Mor1] Charles B. Morrey, Jr. ”The Problem of Plateau on a Riemannian Manifold” Annals of Math., 49, No. 4 (1948), pp. 807-851. [Mul] Müller, Stefan ”Higher integrability of determinants and weak convergence in L 1 ”. J. Reine Angew. Math. 412 (1990), 20–34. [Oss] Osserman,Robert”Aproofoftheregularityeverywhereof the classical solution to Plateau’s problem”. Ann. of Math. (2) 91 1970 550569. [OGR] Gulliver, R. D., II; Osserman, R.; Royden, H. L. ”A theoryof branchedimmersionsofsurfaces”.Amer.J. Math. 95 (1973), 750812. 210
[Poi] Poisson, Siméon Denis ”Extrait d’un mémoire sur les surfaces élastiques” Correspondance de l’École Polytechnique 3, 154 (1816). [Riv] Rivière, Tristan ”Everywhere discontinuous harmonic maps into spheres.” Acta Math. 175 (1995), no. 2, 197226. [Riv1] Rivière, Tristan ”Conservation laws for conformally invariantvariationalproblems.”Invent.Math.168(2007),no. 1, 1–22. [Riv2] Rivière, Tristan ”Analysis aspects of Willmore surfaces,” Inventiones Math., 174 (2008), no.1, 1-45. [Riv3] Rivière, Tristan ”Sub-criticality of Schrödinger Systems with Antisymmetric Potentials”, preprint (2009). [Riv4] Rivière, Tristan ”<strong>Variational</strong> Principles for immersed Surfaces with L2-bounded Second Fundamental Form”, arXiv:1007.2997 (2010). [RiSt] Rivière, Tristan; Struwe, Michael ”Partial regularity for harmonic maps and related problems.” Comm. Pure Appl. Math. 61 (2008), no. 4, 451–463. [Rud] Rudin, Walter ”Real and complex analysis”. Third edition. McGraw-Hill Book Co., New York, 1987. [Sha] Shatah, Jalal ”Weak solutions and development of singularities of the SU(2) σ-model.” Comm. Pure Appl. Math. 41 (1988), no. 4, 459–469. [ShS] Shatah, Jalal; Struwe, Michael The Cauchy problem for wave maps. Int. Math. Res. Not. 2002, no. 11, 555–571. [Tao1] Tao, Terence ”Global regularity of wave maps. I. Small critical Sobolev norm in high dimension.” Internat. Math. Res. Notices 2001, no. 6, 299–328. 211
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 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: larityof weak solutionsof nonlinear
show all

Conformally Invariant Variational Problems. - SAM

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

Delete template?

Save as template?