Francine Meylan

Approximation and convergence of formal CR-mappings

An important step in understanding the existence of analytic objectswith certain properties consists of understanding the same problem at the level of formal power series. The latter problem can be reduced to a sequence of algebraic equations for the coefficients of the unknown power series and is often simpler than the original problem, where the power series are required to be convergent. It is therefore of interest to know whether such power series are automatically convergent or can possibly be replaced by other convergent power series satisfying the same properties. A celebrated result of this kind is Artin’s approximation theorem [1] which states that a formal solution of a system of analytic equations can be replaced by a convergent solution of the same system that approximates the original solution at any prescribed order. In this paper, we study convergence and approximation properties (in the spirit of [1]) of formal (holomorphic) mappings sending real-analytic submanifolds M ⊂ C and M ′ ⊂ C ′ into each other, N,N ′ ≥ 2. In this situation, the above theorem of Artin cannot be applied directly. Moreover, without additional assumptions on the submanifolds, the analogous approximation statement is not even true. Indeed, in view of an example of Moser-Webster [23], there exist real-algebraic surfaces M,M ′ ⊂ C that are formally but not biholomorphically equivalent. However, our firstmain result shows that this phenomenon cannot happen if M is a minimal CR-submanifold (not necessarily algebraic) in C (see Section 2.1 for notation and definitions). Theorem 1.1. Let M ⊂ C be a real-analytic minimal CR-submanifold and M ′ ⊂ C ′ a real-algebraic subset with p ∈ M and p ′ ∈ M ′. Then for any formal (holomorphic)

Degree of a holomorphic map between unit balls from ℂ² to ℂⁿ

Let f be a rational proper holomorphic map between the unit ball in C 2 and the unit ball in C n . Write 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 degf = max{deg(p j ) j = 1,..,n ,degq}. In this paper, we give a bound estimate for the degree of f, improving the bound given by Forstneric (1989).

Holomorphic Approximation in Banach Spaces: A Survey

We give a survey about the Runge approximation problem for a holomorphic function defined on the unit ball of a complex Banach space. More precisely, we ask whether such a holomorphic function can be uniformly approximated on smaller balls by functions that are holomorphic on the entire space. This turns out to be a subtle (open) question, whose (partial) resolution in the past 15 years played a central role in deeper investigations in complex analysis in Banach spaces.

A Regularity Result for CR Mappings Between Infinite Type Hypersurfaces

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 f a holomorphic function defined on one side of M, extending continuously through M, and mapping M into M′. Then f has a holomorphic extension across M. In this paper, we extend this classical theorem to higher complex dimensions for a class of hypersurfaces of infinite type.

Extension de germes de diffeomorphismes cr pour une classe d'hypersurfaces analytiques reelles non essentiellement finies dans

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.