# Francine MeylanUniversité de Fribourg · Department of Mathematics

Francine Meylan

PhD in Mathematics UCSD

Department of Mathematics, University of Fribourg

## About

46

Publications

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326

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## Publications

Publications (46)

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.

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...

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...

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...

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...

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.

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...

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.

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...

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...

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.

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.

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.

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.

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.

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.

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...

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...

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...

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...

We give a survey about the Runge approximation problem for a holomorphic
function defined on the unit ball of a complex Banach space.

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...

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.

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 .

It is shown that a real-valued formal meromorphic function on a formal generic submanifold of finite Kohn-Bloom-Graham type is necessarily constant.

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...

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...

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...

We prove several analyticity results for CR-mappings of positive codimension for which the target is a real-algebraic CR-submanifold.

We survey some recent results on holomorphic or formal mappings sending real submanifolds into each other in complex space.

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...

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...

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...

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.

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.

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...