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Alan Noell

Publications and source records attributed to Alan Noell.

10 recordsLinked to original sources
arXiv2024

CR functions at CR singularities: approximation, extension, and hulls

We study three possible definitions of the notion of CR functions at CR singular points, their extension to a fixed-neighborhood of the singular point, and analogues of the Baouendi--Tr\`eves approximation in a fixed neighborhood. In particular, we give a construction of certain disc hulls, which, if large enough, give the fixed-neighborhood extension and approximation properties. We provide many examples showing the distinctions between the classes and the various properties studied.

math.CV↗
arXiv2021

Cartan uniqueness theorem on nonopen sets

Cartan's uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class of sets such as local real-analytic subvarieties or subanalytic sets, not just submanifolds. Suppose that $V$ is a locally connected and locally closed subset of ${\mathbb{C}}^n$ such that the hull constructed by contracting analytic discs close to arbitrarily small neighborhoods of a point always contains the point in the interior. Then restrictions of holomorphic functions uniquely extend to a fixed neighborhood of the point. Using this extension, we obtain a version of Cartan's uniqueness theorem for such sets. When $V$ is a real-analytic subvariety, we can generalize the concept of infinitesimal CR automorphism and also prove an analogue of the theorem. As an application of these two results we show that, for circular subvarieties satisfying the condition, the only automorphisms, CR or infinitesimal, are linear.

math.CV↗
arXiv2020

On CR singular CR images

We say that a CR singular submanifold $M$ has a removable CR singularity if the CR structure at the CR points of $M$ extends through the singularity as an abstract CR structure on $M$. We study such real-analytic submanifolds, in which case removability is equivalent to $M$ being the image of a generic real-analytic submanifold $N$ under a holomorphic map that is a diffeomorphism of $N$ onto $M$, what we call a CR image. We study the stability of the CR singularity under perturbation, the associated quadratic invariants, and conditions for removability of a CR singularity. A lemma is also proved about perturbing away the zeros of holomorphic functions on CR submanifolds, which could be of independent interest.

math.CV↗
arXiv2019

A CR singular analogue of Severi's theorem

Real-analytic CR functions on real-analytic CR singular submanifolds are not in general restrictions of holomorphic functions, unlike in the CR nonsingular case. We give a simple condition that completely characterizes those quadric CR singular manifolds of codimension 2 in ${\mathbb C}^{n+1}$ for which an extension result holds. Consequently, we obtain an extension result for general real-analytic CR singular submanifolds of codimension 2. As applications we give a condition for the flattening of such submanifolds, and we classify CR singular images of CR submanifolds up to second order.

math.CV↗
arXiv2018

On the Levi-flat Plateau problem

We solve the Levi-flat Plateau problem in the following case. Let $M \subset {\mathbb C}^{n+1}$, $n \geq 2$, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose $M$ is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in ${\mathbb C}^n \times {\mathbb R}$ with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary $M$. We also study boundary regularity of CR automorphisms of domains in ${\mathbb C}^n \times {\mathbb R}$.

math.CV↗
arXiv2017

On Lewy extension for smooth hypersurfaces in ${\mathbb C}^n \times {\mathbb R}$

We prove an analogue of the Lewy extension theorem for a real dimension $2n$ smooth submanifold $M \subset {\mathbb C}^{n}\times {\mathbb R}$, $n \geq 2$. A theorem of Hill and Taiani implies that if $M$ is CR and the Levi-form has a positive eigenvalue restricted to the leaves of ${\mathbb C}^n \times {\mathbb R}$, then every smooth CR function $f$ extends smoothly as a CR function to one side of $M$. If the Levi-form has eigenvalues of both signs, then $f$ extends to a neighborhood of $M$. Our main result concerns CR singular manifolds with a nondegenerate quadratic part $Q$. A smooth CR $f$ extends to one side if the Hermitian part of $Q$ has at least two positive eigenvalues, and $f$ extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed.

math.CV↗
arXiv2017

Extension of CR functions from boundaries in ${\mathbb C}^n \times {\mathbb R}$

Let $Ω\subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial Ω$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial Ω\to {\mathbb C}$ be a smooth function that is CR at CR points of $\partial Ω$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\barΩ$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial Ω$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\barΩ$ in ${\mathbb C}^{n+1}$. An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of $\partial Ω$ as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.

math.CV↗
arXiv2017

Codimension two CR singular submanifolds and extensions of CR functions

Let $M \subset {\mathbb{C}}^{n+1}$, $n \geq 2$, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real-analytic function on $M$ that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of $M$. Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let $Ω\subset {\mathbb{C}}^n \times {\mathbb{R}}$, $n \geq 2$, be a bounded domain with a connected real-analytic boundary such that $\partial Ω$ has only nondegenerate CR singularities. We prove that if $f \colon \partial Ω\to {\mathbb{C}}$ is a real-analytic function that is CR at CR points of $\partial Ω$, then $f$ extends to a holomorphic function on a neighborhood of $\overlineΩ$ in ${\mathbb{C}}^n \times {\mathbb{C}}$.

math.CV↗