**Workshop
« Dynamics and CR geometry », Nice, 26-30th
June 2017.**

**Speakers : **

P. Ebenfelt (UC San Diego) ;

X. Gong (U. Wisconsin-Madison) ;

X. Huang (Rutgers U.) ;

M. Kolar (U. Masaryk, Brno) ;

I. Kossovskiy (U. Masaryk, Brno) ;

B. Lamel (U. Vienna) ;

F. Meylan (U. Fribourg) ;

N. Mir (Texas A&M at Quatar) ;

S. Nivoche (UNSA) ;

R. Shafikov (U. West Ontario) ;

L. Stolovitch (CNRS-UNSA)

D. Zaitsev (Trinity College Dublin)

**Schedule :**

**Monday : **

9h30 : F. Meylan

"Nonlinear symmetries of polynomial models"

10h30-10h45 : coffee break

10:45-11:45 : D. Zaitsev

"Jet vanishing orders and effectivity of Kohn's algorithm in dimension "

14:30 -15:30: X. Gong

"A Frobenius-Nirenberg theorem with parameter"

15:30- : free discussions in groups

**Tuesday: **

9h30 : M. Kolar

"Local symmetries of Levi degenerate hypersurfaces"

10h30-10h45 : coffee break

10:45-11:45 : R. Shafikov

"Discs in hulls of real immersions into Stein manifolds"

14:30 -15:30: L. Stolovitch

« Big denominators and analytic normal forms »

15:30- : free discussions in groups

**Wednesday: **

9h30 : P. Ebenfelt

"The obstruction function and deformations of three-dimensional strictly pseudoconvex CR manifolds."

10h30-10h45 : coffee break

10:45-11:45 : I. Kossovskiy

"A convergent normal form for everywhere Levi degenerate hypersurfaces in C^3"

14:30 -15:30: X. Huang

"On the flattening of a non-degenerate CR singular point on a real-submanifold of real codimension two in C^n."

15:30- : Problems session

**Thursday:**

9h30 : B. Lamel

"Convergence of the Chern-Moser-Beloshapka normal forms"

10h30-10h45 : coffee break

10:45-11:45 : D. Zaitsev

"A geometric approach to Catlin's boundary systems"

14:30 -15:30: N. Mir

"Convergence of formal CR maps between real submanifolds in complex space"

15:30- : free discussions in groups

**Friday: **

9h30 : S. Nivoche

"A new proof of a problem of Kolmogorov on $\epsilon$-entropy. "

10h30-10h45 : coffee break

10:45-11:45 : X. Gong

" Holder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary"

**Abstracts :
**

*P. Ebenfelt : The obstruction function and deformations of three-dimensional strictly pseudoconvex CR manifolds.

Abstract: The obstruction function on the (smooth) boundary of a strictly pseudoconvex domain in $\mathbb C^n$ arises as the obstruction to smoothness up to the boundary of the Cheng-Yau solution to a particular Dirichlet problem. Graham showed that the obstruction function is actually a local CR invariant on the boundary. In $\mathbb C^2$, this function occurs also as the lowest order term in the Bergman kernel. Moreover, it arises in variational formulas for global invariants such as Q’-curvature and the Burns-Epstein invariant under deformations. In this talk, we will explain this, and discuss what can be gleaned from the vanishing of the obstruction function.

*X. Gong : Holder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary

Abstract: I will discuss a homotopy formula for a strictly pseudoconvex domain with C^2 boundary. This will be used to obtain estimate gaining one half derivative in the Holder-Zygmund space. The estimate is applied for the boundary regularities of d-bar solutions for strictly pseudoconvex domains and D-solutions for a suitable domain in the Levi-flat Euclidean space.

*X. Gong : A Frobenius-Nirenberg theorem with parameter.

Abstract: The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space.We will show two results about the Newlander-Nirenberg theoremwith parameter. The first extends the Newlander-Nirenberg theorem to a parametric version,and its proof yields a sharp regularity result as Webster's proof for the Newlander-Nirenberg theorem. The second concerns a version of Nirenberg's complex Frobenius theorem and its proof yields a result with a mild loss of regularity.

*X. Huang : On the flattening of a non-degenerate CR singular point on a real-submanifold of real codimension two in C^n.

Abstract: This is a joint work with Hanlong Fang. I will discuss when a real analytic submanifold of real codimension two can be mapped into the standard Levi-flat hypersurface C^{n-1}\times R near a non-degenerate CR singular point. Surprisingly, we found that this can always be done except that both directions are parabolic. We will describe a proof of this and also present many examples showing that the results are arguments are more or less the best possible ones.

*M. Kolar : Local symmetries of Levi degenerate hypersurfaces

Abstract:
We will discuss some recent results concerning classification of

Levi degenerate hypersurfaces according to their Lie algebra of infinitesimal CR automorphisms.

*I. Kossovskiy : A convergent normal form for everywhere Levi degenerate hypersurfaces in C^3

Abstract: The existence of everywhere Levi degenerate hypersurfaces in C^n (n\geq 3) having finite dimensional automorphism groups is among the most intriguing discoveries in CR-geometry in the 90's. In the case n=3, homogeneous structures of the latter kind lead to the class of everywhere 2-nondegenerate hypersurfaces. This class, among other things, plays an important role in the geometry of bounded symmetric domains, that is why its geometry has been studied intensively for the last 20 years. A differential-geometric solution for the CR equivalence problem for this class of hypersurfaces was given in the work of Isaev and Zaitsev. However, the power series approach to the equivalence problem (namely, a complete normal form) has not been realized yet, the main difficulty being the absence of polynomial models for the class of everywhere Levi degenerate hypersurfaces in C^3. On the language of Catlin's boundary invariants theory, this difficulty reads as the infinite multitype of the latter class of hypersurfaces.

In our joint work with Martin Kolar, we give a solution to the above described problem, by presenting a complete convergent normal form for the class of everywhere Levi degenerate hypersurfaces in C^3. Our normal form is defined up to the automorphism group of the model manifold: the tube over the future light cone in the 3-dimensional time-space. The key tool that we use for constructing a normal form is a homological process, based on a non-polynomial model (the above tube over the future light cone). As an application of the normal form, we give a complete list of possible dimensions for the isotropy group of an everywhere Levi degenerate hypersurfaces in C^3.

*B. Lamel : Convergence of the Chern-Moser-Beloshapka normal forms

Abstract : In this talk, we first describe a normal form of real-analytic, Levi-nondegenerate submanifolds of $\CN$ of codimension $d\geq 1$ under the action of formal biholomorphisms, that is, of perturbations of Levi-nondegenerate hyperquadrics. We give a sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. We show that our techniques can be adapted in the case $d=1$ in order to obtain a new and direct proof of Chern-Moser normal form theorem.

*N. Mir : Convergence of formal CR maps between real submanifolds in complex space

Abstract: We will discuss our recent joint work with B. Lamel establishing the convergence of formal mappings between real-analytic strongly pseudoconvex CR manifolds in complex spaces of different dimension.

*F. Meylan : Nonlinear symmetries of polynomial models

Abstract : In this talk, we will discuss infinitesimal CR automorphisms of Levidegenerate hypersurfaces. We illustrate the recent general resultsof \cite{KMZ}, \cite{KM1}, \cite{FM2}, on a class of concreteexamples, polynomial models in $\mathbb C^3$ of the form $\Im\; w= \Re(P(z) \overline{Q(z)}) $, where $P$ and $Q$ are weighted homogeneous holomorphic polynomials. We classify such modelsaccording to their Lie algebra of infinitesimal CR automorphisms.

This is a joint work with Martin Kolar.

*S.
Nivoche : A new proof of a problem of Kolmogorov on
$\epsilon$-entropy.

Abstract : In the 80's, a Kolmogorov
problem about the $\epsilon$-entropy of a class of analytic function
was stated : $$\lim_{\epsilon \to 0}
\frac{H_\epsilon(A_K^D}{\ln^{n+1}(1/\epsilon)} =
\frac{2C(K,D)}{(2\pi)^n(n+1)!}.$$

In 04, this problem was solved
by using technics of pluripotential theory and in particular by
proving a Conjecture of Zakharyuta.

Here we will present a new
proof of Kolmogorov's problem, independently of this conjecture. We
will use the asymptotic behaviour of the Bergman kernel of a
concentration operator and some properties of special analytic
polyhedra. It is a common work with Oscar Bandtlow (London).

*R. Shafikov : Discs in hulls of real immersions into Stein manifolds

Abstract: Gromov’s classical result states that a compact Lagrangian

submanifold of C^n admits a nonconstant holomorphic disc attached

to it. I will discuss partial generalizations of this result to immersions

into Stein manifolds. This is joint work with A. Sukhov.

*L. Stolovitch : Big denominators and analytic normal forms

Abstract : We study the regular action of an analytic pseudo-group of transformations on the space of germs of various analytic objects of local analysis and local differential geometry. We fix a homogeneous object F_0 and we are interested in an analytic normal form for the whole affine space F_0 + h.o.t. We prove that if the cohomological operator defined by F_0 has the big denominators property and if the sapce of formal normal forms is well chosen then this formal normal form holds in analytic category.

We also define big denominators in systems of nonlinear PDEs and prove a theorem on local analytic solvability of systems of nonlinear PDEs with big denominators. This result can be used in several problems, in particular the one to be presented by B. Lamel.

*D. Zaitsev : Jet vanishing orders and effectivity of Kohn's algorithm in dimension 3

Abstract : We propose a new class of geometric invariants called jet vanishing

orders, and use them to establish a new selection algorithm for

generators of the Kohn's multiplier ideals for special domains in

dimension 3. In particular, we obtain the effective termination of our

selection algorithm with explicit bounds.

*D. Zaitsev: A geometric approach to Catlin's boundary systems

Abstract : For a point p in a smooth real hypersurface $M\subset\C^n$, where the

Levi form has the nontrivial kernel K10p, we introduce an invariant

cubic tensor $\tau^3_p \colon \C T_p \times K^{10}_p \times

\overline{K^{10}_p} \to \C\otimes (T_p/H_p)$, which together with

Ebenfelt's tensor ψ3, constitutes the full set of 3rd order invariants

of M at p.

Next, in addition, assume $M\subset\C^n$ to be {\em (weakly)

pseudoconvex}. Then τ3p must identically vanish. In this case we

further define an invariant quartic tensor $\tau^4_p \colon \C T_p

\times \C T_p \times K^{10}_p\times \overline{K^{10}_p} \to \C\otimes

(T_p/H_p)$, and for every q=0,…,n−1, an invariant submodule sheaf of

(1,0) vector fields in terms of the Levi form, and an invariant ideal

sheaf of complex functions generated by certain derivatives of the

Levi form, such that the set of points of Levi rank q is locally

contained in certain real submanifolds defined by real parts of the

functions in the ideal sheaf, whose tangent spaces have explicit

algebraic description in terms of the quartic tensor τ4.

Finally, we relate the introduced invariants with D'Angelo's finite

type, Catlin's mutlitype and Catlin's boundary systems.