Cauchy-Riemann (CR) Manifolds, Szeg o kernel asymptotics and Morse - PowerPoint PPT Presentation

Cauchy-Riemann (CR) Manifolds, Szeg¨ o kernel asymptotics and Morse inequalities on CR-manifolds This presentation will about the following paper which constitutes my m´ emoire for M2: C.Y. Hsiao, G. Marinescu. Szeg¨ o kernel asymptotics and Morse inequalities on CR manifolds

Outline 1. Overview of the theory of Cauchy Riemann manifolds 2. Sub-elliptic estimates for the Kohn Laplacian 3. Szeg¨ o kernel, Maximal function, and Weak Morse Inequalities 4. Strong Morse Inequalities

Overview of the theory of Cauchy-Riemann manifolds Let X be a manifold of real dimension 2 n − 1 for n � 2. We consider its tangent bundle TX and its complexification: C TX := TX ⊗ R C . We suppose that T 1 , 0 X is a complex sub-bundle of the complexified tangent bundle C TX . We say that X is Cauchy-Riemann if it satisfies the following properties 1. The complex rank of the tangent bundle T 1 , 0 X is n − 1. 2. T 1 , 0 X ∩ T 0 , 1 X = { 0 } where T 0 , 1 X = T 1 , 0 X . 3. Let U be an open set. For any V 1 , V 2 in C ∞ ( U , T 1 , 0 X ), we have [ V 1 , V 2 ] ∈ C ∞ ( U , T 1 , 0 X ).

Similarities and Differences? What are the similarities and differences between the things that are done in Cauchy-Riemann manifolds, and the things that are done in complex manifolds? Both have the differential operator d defined locally on smooth ( p , q )-forms as follows:   � f IJ dz I ∧ dz J d   | I | = p , | J | = q �� � ∂ f IJ dz i ∧ dz I ∧ dz J + ∂ f IJ � dz i ∧ dz I ∧ dz J = . ∂ z i ∂ z i | I | = p , | J | = q i

Similarities and Differences? For complex manifolds: ∂ maps df to � ∂ f IJ ∂ z i dz i ∧ dz I ∧ dz J . For Cauchy-Riemann manifolds, ∂ b is defined by the following composition: � ∧ p +1 , q T ∗ X ⊕ ∧ p , q +1 T ∗ X π p , q +1 � ∧ p , q +1 T ∗ X . d ∧ p , q T ∗ X It is not clear that for Cauchy-Riemann manifolds, ∂ b ◦ ∂ b = 0. Turns out to be true thanks to Cartan-Lie formula.

Is ∂ b ◦ ∂ b = 0? Cartan-Lie Formula gives the following lemma: Lemma Let q � 1 and ω be a smooth (0 , q )-form. For all x ∈ X , if ω ( x ) annihilates ∧ 0 , q T x X , then d ω ( x ) annihilates ∧ 0 , q +1 T x X . Then for v ∈ ∧ 0 , q T x X , we have df ( v ) = π 0 , q +1 df ( v ) so that (1 − π 0 , q +1 ) df = 0. Hence by previous lemma, we have d (1 − π 0 , q +1 ) df = 0. Therefore, π 0 , q +2 d ◦ π 0 , q +1 df ∂ b ◦ ∂ b f = π 0 , q +2 d ◦ π 0 , q +1 df + π 0 , q +2 d (1 − π 0 , q +1 ) df = � �� � =0 π 0 , q +2 d 2 f = 0 . =

Complexes Let Ω p , q ( X ) be a space of smooth ( p , q )-forms. And thus we have a complex. ∂ b ∂ b � Ω p , 0 ( X ) � ... � Ω p , n − 1 ( X ) � 0 0 Let ( X ) = ker ∂ b : Ω p , k ( X ) → Ω p , k +1 ( X ) H p , k im ∂ b : Ω p , k − 1 ( X ) → Ω p , k ( X ) . b Now given an inner product ( − , − ) acting on C TX , we may also ∗ have ∂ b . Let ∗ b ) 2 � b = ( ∂ b + ∂ be the Kohn Laplacian.

Hodge I The second similarity and difference is that Hodge theory applies for both cases, except the ways to prove them are different. For Dolbeault complexes on complex manifolds, the first part of the Hodge theory says that H p , k ( X ) is finite-dimensional. ∂ But we see that the cohomology group has a finite-dimension because the kernel of the Laplacian is finite-dimensional. This is due to the Hodge isomorphism. To prove this, we ask a simple question: when is a vector subspace of a Hilbert space finite-dimensional? Sufficient condition: when every bounded sequence has a convergent subsequence. (Also known as Riesz’s theorem.)

Hodge I The tool used here is the inclusion of Sobolev spaces: W ∞ ( p , q ) ( X ) = Ω p , q ( X ) ⊆ ... ⊆ W 2 ( p , q ) ( X ) ⊆ W 1 ( p , q ) ( X ) ⊆ L 2 ( p , q ) ( X ) . Observation: G˚ arding’s inequality + Sobolev’s theorem implies that solutions to ∆ u = 0 are always smooth, following from the fact that there exists k > 0 such that � u � 2 m + k � C ( � u � 2 L 2 + � ∆ u � 2 m ) = C � u � 2 L 2 for each m (in the weak sense). By Sobolev’s theorem, u is a smooth form. Let { u k } k be a bounded sequence in ker ∆, which is seen as a vector subspace of L 2 ( X ). For each l , using G˚ arding’s inequality, we have � u k � l � C � u k � L 2 .

Hodge I Therefore, the sequence { u k } is uniformly bounded by some constant M in W l ( p , q ) ( X ) for each l . This means that { u k } is uniformly bounded in W 1 ( p , q ) ( X ). By Rellich’s lemma, it has a convergent subsequence in L 2 ( p , q ) ( X ), say { u k i 1 } Using this subsequence, it is uniformly bounded in W 2 ( p , q ) ( X ), hence we may extract a subsequence of this subsequence which converges in W 1 ( p , q ) ( X ). Continuing this way, we extract the diagonal elements. We then obtain a subsequence of { u k } that converges in W l ( p , q ) ( X ) for all l . By Sobolev’s theorem, the limit of this converging subsequence is smooth.

Hodge I Let the limit be u = lim u k . For all v ∈ L 2 ( p , q ) ( X ), we have 0 = � ∆ u k , v � = � u k , ∆ v � , where ∆ v is defined in the distributional sense. We may in fact avoid this by choosing v = ∆ u straightaway. Since strong convergence implies weak convergence, we have � ∆ u , ∆ u � = � u , ∆∆ u � = lim k →∞ � u k , ∆∆ u � = lim k →∞ � ∆ u k , ∆ u � = 0 . Hence ∆ u = 0, and ker ∆ is finite dimensional. Hence G˚ arding’s inequality is the essential ingredient for Hodge theory.

Hodge II The second part of the Hodge theory says that H p , q ∂ ( X ) is isomorphic to the kernel of the laplacian restricted to the smooth ( p , q ) forms. The following ingredients are used for Hodge theory: ∗ ) 2 , 1. The Laplacian ∆ = ( ∂ + ∂ 2. The existence of the Green’s function (which is well-defined by Hodge I) G on the eigenspaces of ∆ with eigenvalues � = 0 such that G ∆ = ∆ G = id . Extend G by zero on the kernel of ∆. This implies that ∂ ◦ G = G ◦ ∂ . Then we have ∆ ◦ G = id − proj ker ∆ . This implies that at the level of Dolbeault cohomology, id = proj ker ∆ .

Hodge II Why? For α ∈ H p , q ∂ ( X ), we have ∗ G + ∂ ∗ ∂ G ) α = 0 (In cohomology) + ∂ ∗ G ∂α = 0 . ( ∂∂ Hence id − proj ker ∆ = 0 at the level of cohomology. This means that for each α ∈ ker ∂ , α − proj ker ∆ α = ∂β for some β ∈ H p , q − 1 ( X ). We have therefore an important ∂ decomposition: ker ∂ = im ∂ ⊕ ker ∆ . Taking quotient by the image, we get the second important Hodge result: H p , k ( X ) ∼ = ker ∆ . ∂

Hodge (Summary) Therefore, G˚ arding’s inequality is essential. G˚ arding’s inequality + Sobolev’s theorem + Rellich lemma implies the finiteness of dimension of the space of harmonic functions. This implies well-definedness of the Green’s function, and hence the equivalences between the cohomology group and the space of harmonic functions (link between algebra and analysis). Do we have G˚ arding’s inequalities in the case of Cauchy-Riemann manifolds?

Sub-elliptic Estimates of the Kohn Laplacian

Sub-elliptic Estimates

Sub-elliptic Estimates of the Kohn Laplacian We may not have the G˚ arding’s inequality for the Cauchy-Riemann manifolds, but we may obtain something weaker for the Kohn Laplacian, which still implies the regularity theorem. The ingredients for this to work are: 1. The Levi form. 2. The eigenvalues of the Levi form (or we call it later the Y ( q ) condition). 3. Elliptic regularisation techniques.

The Levi Forms and the Y ( q ) condition We recall that the complexified tangent bundle of X may be written as C TX = T 1 , 0 X ⊕ T 0 , 1 X ⊕ C T . Let w 0 be a 1-form dual to T . Here T is the vector field which is transversal to T 1 , 0 X and T 0 , 1 X . Definition of Levi Forms : Let p ∈ X be given. Let Z and W be elements in ∧ 1 , 0 T p X , and ˜ Z and ˜ W be smooth vector fields C ∞ ( X , ∧ 1 , 0 TX ) such that ˜ Z ( p ) = Z and ˜ W ( p ) = W . The Levi form L p is the Hermitian quadratic form defined as 1 2 √− 1 � [˜ Z , ˜ L p ( Z , W ) := W ]( p ) , w 0 ( p ) � . By Cartan-Lie formula, this is independent of the choice of ˜ Z and ˜ W whose evaluation at p are respectively Z and W . At the point p , the Levi forms may be written as a matrix. Therefore, it makes sense to talk of its eigenvalues.

The Levi Forms and the Y ( q ) condition Definition of Y ( q ) condition : The Levi form acting on ( p , q )-forms is said to satisfy the Y ( q ) condition if there are max( n − q , q + 1) eigenvalues of the same sign, or min( n − q , q + 1) pairs of eigenvalues of the different signs. This leads to an important result: let { L k } be a set of local orthonormal frames of T 1 , 0 X with its corresponding dual ω k . Proposition : Let X be a compact oriented Cauchy-Riemann manifold, and φ be a ( p , q )-form with compact support. Let φ IJ be ω I ∧ ω J component of φ , then L k + | Re( Y ( φ IJ ) , φ IJ ) | � � � b φ � 2 + � φ � 2 := Q b ( φ, φ ) , � φ � 2 L K + � φ � 2 where n − 1 � � φ � 2 � L k φ � 2 . L k = k =1