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Christine Laurent-Thiébaut

Publications and source records attributed to Christine Laurent-Thiébaut.

12 recordsLinked to original sources
arXiv2020

Holomorphic Approximation via Dolbeault Cohomology

The purpose of this paper is to study holomorphic approximation and approximation of $\bar\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the smooth and the $L^2$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.

math.CV↗
arXiv2020

Holomorphic Approximation and Mixed Boundary Value Problems for $\partial$

In this paper we study holomorphic approximation using boundary value problems for $\bar\partial$ on an annulus in the Hilbert space setting. The associated boundary conditions for $\bar\partial$ are the mixed boundary problems on an annulus. We characterize pseudoconvexity and Runge type property of the domain by the vanishing of related $L^2$ cohomology groups.

math.CV↗
arXiv2017

Hearing pseudoconvexity in Lipschitz domains with holes via $\overline\partial$

Let $Ω=\widetildeΩ\setminus \overline{D}$ where $\widetildeΩ$ is a bounded domain with connected complement in $\mathbb C^n$ (or more generally in a Stein manifold) and $D$ is relatively compact open subset of $\widetildeΩ$ with connected complement in $\widetildeΩ$. We obtain characterizations of pseudoconvexity of $\widetildeΩ$ and $D$ through the vanishing or Hausdorff property of the Dolbeault cohomology groups on various function spaces. In particular, we show that if the boundaries of $\widetildeΩ$ and $D$ are Lipschitz and $C^2$-smooth respectively, then both $\widetildeΩ$ and $D$ are pseudoconvex if and only if $0$ is not in the spectrum of the $\overline\partial$-Neumann Laplacian on $(0, q)$-forms for $1\le q\le n-2$ when $n\geq 3$; or $0$ is not a limit point of the spectrum of the $\overline\partial$-Neumannn Laplacian on $(0, 1)$-forms when $n=2$.

math.CV↗
arXiv2016

Solving $\overline\partial$ with prescribed support on Hartogs triangles in $\mathbb C^2$ and $\mathbb C\mathbb P^2$

In this paper we study the solvability of the Cauchy-Riemann equation with prescibed support in different spaces of forms. The unbounded Hartogs triangle in $\mathbb C^2$ and the Hartogs domains in $\mathbb C\mathbb P^2$ provide us new unexpected phenomena. In particular we prove that the Dolbeault isomorphism fails to hold for the Dolbeault cohomology with prescribed support

math.CV↗
arXiv2016

On the $L^2$-Dolbeault cohomology of annuli

For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\partial}$-equation on the non-smooth domains which are the holes of these annuli.

math.CV↗
arXiv2013

$L^p$-theory for the tangential Cauchy-Riemann equation

We are interested in $L^p$-theory for the tangential Cauchy-Riemann operator in locally embeddable, $s$-concave, generic CR manifolds. We study the Dolbeault isomorphism and develop the Andreotti-Grauert theory in that setting. Using Serre duality, we solve the tangential Cauchy-Riemann equation with exact support and $L^p$-estimates.

math.CV↗
arXiv2013

Théorie $L^p$ pour l'équation de Cauchy-Riemann

In this paper we propose a systematic study of the Cauchy-Riemann operator in the $L^p$-setting in complex manifolds. We first consider $L^p_{loc}$-theory and then we develop an $L^p$ Andreotti-Grauert theory. Finally we consider Serre duality and its applications to the solvability of the Cauchy-Riemann equation with exact support in $L^p$-spaces.

math.CV↗
arXiv2008

Stability of the vanishing of the $\overline\partial_b$-cohomology under small horizontal perturbations of the CR structure in compact abstract q-concave CR manifolds

We consider perturbations of CR structures which preserve the complex tangent bundle. For a compact generic CR manifold its concavity properties and hence the finiteness of some $\overline\partial_b$-cohomology groups are also preserved by such perturbations of the CR structure. Here we study the stability of the vanishing of these groups.

math.CV↗