Conformally Invariant Variational Problems. - SAM

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I Introduction These lecture notes form the cornerstone between two areas of Mathematics: calculus of variations and conformal invariance theory. Conformalinvarianceplaysasignificantrole in many areasof Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian surfaces, Weyl tensors, Q-curvature, Yang-Mills fields, etc... We shall be concerned with the study of conformal invariance in analysis. More precisely, we will focus on the study of nonlinear PDEs arising from conformally invariant variational problems (e.g. harmonic maps, prescribed mean curvature surfaces, Yang-Mills equations, amongst others). 2
II Conformal transformations - some fundamental results. Conformal invariance is a universal property. A transformation is called conformal when it preserves angles infinitesimally, that is, when its differential is a similarity at every point. Unlike in higher dimensions, the group of conformal transformations in two dimensions is very large ; it has infinite dimension. In fact, it contains as many elements as there are holomorphic and antiholomorphic maps. This particularly rich feature motivates us to restrict our attention on the two-dimensional case. Although we shall not be concerned with higher dimension, the reader should know that many of the results presented in these notes can be generalized to any dimension. Definition II.1. A C 1 map u between two riemannian manifolds (M m ,g) and (N n ,h) is said to be conformal if at every point p ∈ M m du p is a composition of isometries from T p M m into du p (T p M m ) and dilations. This is equivalent to ∀p ∈ M m ∀X,Y ∈ T p M m < du p ·X,du p ·Y > h |X| g |Y| g = |du p ·X| h |du p ·Y| h < X,Y > g (II.1) Where < ·,· > g and < ·,· > h denotes respectively the scalar products g on T p M m and h on du p (T p M) C T u(p) N n . ✷ Lemma II.1. A map u is conformal if and only if there exists a function λ on M m such that ∀p ∈ M m ∀X,Y ∈ T p M m (II.2) < du p ·X,du p ·Y > h = e 2λ(p) < X,Y > g λ(p) is called the conformal factor at p. ✷ This result comes from the following Lemma in linear algebra. 3
Page 1: Conformally Invariant Variational P
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
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acting on maps u ∈ W 1,2 (D 2 ,N
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Whence, [ ∆u−H(u)(∇ ⊥ u,∇
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VII Integrabilitybycompensationtheo
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Accordingly, if φ lies in L ∞ ,
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Proof of the regularity of the solu
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If wemultiplytheLaplaceequationthro
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where u still denotes the normal un
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frames, thereby compensating for th
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tangent frame field to T 2 . Define
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egularity. Note that (VI.26) is equ
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Just as in the proof of the regular
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VIII A proof of Heinz-Hildebrandt
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maps, namely ∑n+1 ( −∆u i =
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Theorem VIII.2. [Riv1] Let N n be a
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If A is almost everywhere invertibl
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the whole regularity result stated
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Bringing altogether (VIII.15), (VII
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In the simpler case when Ω is dive
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Theorem VIII.5. [Uhl], [Riv1] Let m
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yields the existence of the solutio
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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.
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- General relativity : The Willmore
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where ⋆ is the Hodge operator 37
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Let us take locally about p a norma
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and that we have denoted by ( ⃗
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and similarly the second fundamenta
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Let (⃗n 1 ,··· ,⃗n m−2 ) b
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orthonormal basis for the metric g.
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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
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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
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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
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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 ⎧ ⎨
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Combining this inequality with (X.1
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ii) iii) iv) limsupArea( Φ ⃗ k (
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The assumption i) implies that, mod
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Theorem X.14. Let ⃗ Φ be a Lipsc
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Thus < ∇ ⃗ Φ,∇ ⊥ ⃗ L >=
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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
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We have using (X.113) ∂ z ∂ z
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References [Ad] Adams, David R. ”
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[DHKW1] Dierkes, Ulrich; Hildebrand
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larityof weak solutionsof nonlinear
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[Poi] Poisson, Siméon Denis ”Ext
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