Francine Meylan

Francine Meylan
Université de Fribourg · Department of Mathematics

PhD in Mathematics UCSD
Department of Mathematics, University of Fribourg

About

46
Publications
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326
Citations

Publications

Publications (46)
Article
Full-text available
We study the infinitesimal CR automorphisms of polynomial model hypersurfaces of finite multitype, which violates 2-jet determination. We give an exposition of some recent results, which provide explicit description of such “exotic” symmetries in complex dimension three. The results are illustrated by numerous examples.
Article
Full-text available
The existence of a non-defective stationary disc attached to a non-degenerate model quadric in CN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^N$$\end{d...
Article
In this partly expository paper, we deal with sharp jet determination results following from a generalization of the Chern—Moser theory to Levi degenerate hypersurfaces with polynomial models, as obtained in [30]. We formulate the jet determination results for finitely smooth hypersurfaces of finite type. Another goal of the paper is to gain more u...
Preprint
The existence of a nondefective stationary disc attached to a nondegenerate model quadric in C^N is a necessary condition to ensure the unique 1-jet determination of the lifts of a key family of stationary discs. In this paper, we give an elementary proof of the equivalence when the model quadric is strongly pseudoconvex, recovering a result of Tum...
Preprint
We give an explicit construction of a key family of stationary discs attached to a nondegenerate model quadric in $\mathbb{C}^N$ and derive a necessary condition for which (each lift) of those stationary discs is uniquely determined by its $1$-jet at a given point via a local diffeomorphism. This unique $1$-jet determination is a crucial step to de...
Article
We previously introduced a new notion of non-degeneracy for generic real submanifolds in CN. The definition is however not complete and the purpose of this addendum is to complete it.
Preprint
We first construct a counterexample of a generic quadratic submanifold of codimension $5$ in $\Bbb C^9$ which admits a real analytic infinitesimal CR automorphism with homogeneous polynomial coefficients of degree $4.$ This example also resolves a question in the Tanaka prolongation theory that was open for more than 50 years. Then we give sufficie...
Article
We discuss the links between stationary discs, the defect of analytic discs, and 2-jet determination of CR automorphisms of generic nondegenerate real submanifolds of CN of class C4.
Article
Full-text available
We compare various definitions of nondegeneracy of the Levi map for real submanifolds of higher codimension in CN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathb...
Preprint
We classify polynomial models for real hypersurfaces in $\mathbb C^N$, which admit nonlinearizable infinitesimal CR automorphisms. As a consequence, this provides an optimal 1-jet determination result in the general case. Further we prove that such automorphisms arise from one common source, by pulling back via a holomorphic mapping a suitable symm...
Preprint
One constructs an example of a generic quadratic submanifold of codimension $5$ in $\Bbb C^9$ which admits a real analytic infinitesimal CR automorphism with homogeneous polynomial coefficients of degree $3.
Preprint
We discuss the links between stationary discs, the defect of analytic discs, and 2-jet determination of CR automorphisms of generic nondegenerate real submanifolds of C^N of class C^4.
Article
Let M⊂CN be a generic real submanifold of class C⁴. In case M is Levi non-degenerate in the sense Tumanov, we construct stationary discs for M. If furthermore M satisfies an additional non-degeneracy condition, we apply the method of stationary discs to obtain 2-jet determination of CR automorphisms of M.
Preprint
Full-text available
In case M is Levi non-degenerate in the sense Tumanov, we construct stationary discs for $M$. If furthermore M satisfies an additional non-degeneracy condition, we apply the method of stationary discs to obtain 2-jet determination of CR automorphisms of M.
Article
We extend the Chern-Moser approach for hypersurfaces to real submanifolds of higher codimension in complex space to derive results on jet determination for their automorphism group.
Article
We compare various definitions of nondegeneracy for real submanifolds of higher codimension in $\mathbb{C}^N$, and explain why the definition introduced by Beloshapka seems the most relevant for us for finite jet determination problems.
Article
We give a complete classification of polynomial models for smooth real hypersurfaces of finite Catlin multitype in $\mathbb C^3$, which admit nonlinear infinitesimal CR automorphisms. As a consequence, we obtain a sharp 1-jet determination result for any smooth hypersurface with such model. The results also prove a conjecture of the first author ab...
Article
In this paper we study infinitesimal CR automorphisms of Levi degenerate hypersurfaces. We illustrate the recent general results of [18], [17], [15], on a class of concrete examples, polynomial models in ℂ³ of the form Im w = Re (P(z)Q(z)), where P and Q are weighted homogeneous holomorphic polynomials in z = (z1, z2). We classify such models accor...
Article
We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By recent results of Kolar, Meylan and Zaitsev, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear vector fields of arbitrarily high degree, which has no analog in the classical Levi nondegenerate case, or in...
Article
Full-text available
We consider the fundamental invariant of a real hypersurface in CNCN – its holomorphic symmetry group – and analyze its structure at a point of degenerate Levi form. Generalizing the Chern–Moser operator to hypersurfaces of finite multitype, we compute the Lie algebra of infinitesimal symmetries of the model and provide explicit description for eac...
Article
We give a survey about the Runge approximation problem for a holomorphic function defined on the unit ball of a complex Banach space.
Article
We construct an example of a rational mapping from the two-dimensional complex linear space to the complex projective plane, which has indeterminacies on the unit sphere, but such that the image of the unit sphere under this map is contained in the affine part of the complex projective plane and doesn't contain any germ of a non-constant complex cu...
Article
We study the Chern-Moser operator for hypersurfaces of finite type in C 2 . Analysing its kernel, we derive explicit results on jet determination for the stability group, and give a description of infinitesimal CR automorphisms of such manifolds.
Article
Full-text available
Let (V, ξ) be a contact manifold and let J be a strictly pseudoconvex CR structure of hypersurface type on (V, ξ); starting only from these data, we define and we investigate a Differential Graded Lie Algebra which governs the deformation theory of J .
Article
Full-text available
It is shown that a real-valued formal meromorphic function on a formal generic submanifold of finite Kohn-Bloom-Graham type is necessarily constant.
Article
The Schwarz reflection principle in one complex variable can be stated as follows. Let M and M be two real analytic curves in and f a holomorphic function defined on one side of MM extending continuously through MM and mapping M into M Then f has a holomorphic extension across MM In this paper, we extend this classical theorem to higher complex dim...
Article
Let f be a rational proper holomorphic map between the unit ball in C2 and the unit ball in Cn. Write $f = {\frac{(p_1, ..., p_n)}{q}}$ , where pj, j = 1, ..., n, and q are holomorphic polynomials, with $(p_1, ..., p_n, q) = 1$ . Recall that the degree of f is defined by $\deg f = \max \{\deg (p_j)_{j = 1, ..., n}, \deg q\}$ . In this paper, we giv...
Article
Let X be a complex Banach space. Recall that X admits a finite-dimensional Schauder decomposition if there exists a sequence {X n } ∞ n=1 of finite-dimensional subspaces of X, such that every x ∈ X has a unique representation of the form x = ∞ n=1 x n , with x n ∈ X n for every n. The finite-dimensional Schauder decomposition is said to be uncondit...
Article
Full-text available
We prove several analyticity results for CR-mappings of positive codimension for which the target is a real-algebraic CR-submanifold.
Article
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We survey some recent results on holomorphic or formal mappings sending real submanifolds into each other in complex space.
Article
Let M subset of C-N be a minimal real-analytic CR-submanifold and M' subset of C-N' a real-algebraic subset through points p is an element of M and P' is an element of M' respectively. We show that that any formal (holomorphic) mapping f: (C-N, p) --> (C-N', p'), sending M into M', can be approximated up to any given order at p by a convergent map...
Article
Full-text available
Let $M\subset C^N$ be a minimal real-analytic CR-submanifold and $M'\subset C^{N'}$ a real-algebraic subset through points $p\in M$ and $p'\in M'$. We show that that any formal (holomorphic) mapping $f\colon (C^N,p)\to (C^{N'},p')$, sending $M$ into $M'$, can be approximated up to any given order at $p$ by a convergent map sending $M$ into $M'$. If...
Article
Full-text available
In this paper we give general conditions that guarantee the analyticity of ${\mathcal C}^\infty$-smooth CR-mappings between real-analytic CR-submanifolds $M\subset\C^N$ and $M'\subset\C^{N'}$. The proof of the analyticity is based on results on holomorphic and meromorphic extension of functions on wedge-like domains to boundary points that may be o...
Article
Let H:M → M′ be a germ of smooth CR diffeomorphism between M and M′, two real analytic hypersurfaces at 0 in , with M′ given by , where ψ is a real analytic function in a neighborhood of 0 in , satisfying , for some for every choice We prove that H is analytic.
Article
In this article, we prove that smooth CR dieomorphisms between two real analytic holomorphically nondegenerate hypersurfaces, one of which is rigid and polynomial, extend to be locally biholomorphic. It turns out that the result can be generalized to not totally degenerate mappings, in the sense of Baouendi and Rothschild.
Article
The Schwarz reflection principle in one complex variable can be stated as follows. Let M and M’ be two real analytic curves in C and H a holomorphic function defined on one side of M, extending continuously through M, and mapping M into M’. Then H has a holomorphic extension across M. We address here the question of extending this classical theorem...

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