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Erlend F. Wold

Publications and source records attributed to Erlend F. Wold.

8 recordsLinked to original sources
arXiv2026

The polynomially convex embedding dimension of real manifolds of dimension $\leq 11$

We show that any compact smooth real $n$-dimensional manifold $M$ with $n\leq 11$ can be smoothly embedded into $\mathbb{C}^{n+1}$ as a polynomially convex set. In general, there is no such embedding into $\mathbb{C}^n$. This solves a problem by Izzo and Stout for $n\leq 11$. Additionally, we show that the image $\widetilde{M}$ of $M$ in $\mathbb{C}^{n+1}$ is stratified totally real. As a consequence, by a result in [13], each continuous complex-valued functions on $\widetilde{M}$ is the uniform limit on $\widetilde{M}$ of holomorphic polynomials in $\mathbb{C}^{n+1}$. Our proof is based on the jet transversality theorem and a slight improvement of a perturbation result by the first and the third author.

math.CV↗
arXiv2025

Polynomial convexity of $\bar\partial$-flat perturbations of totally real sets

We show that if $X$ is a totally real $d$-dimensional manifold attached to a polynomially convex compact set $K$ in $\mathbb{C}^n$, $d<n$, then there are arbitrarily small perturbations $X'$ of $X$ such that $K\cup X'$ is polynomially convex. The perturbations are induced by diffeomorphisms of $\mathbb{C}^n$ fixing $K$, which are $\bar\partial$-flat on $K\cup X$, and which are arbitrarily $C^k$-close to the identity.

math.CV↗
arXiv2023

Families of Proper Holomorphic Embeddings and Carleman-type Theorems with parameters

We solve the problem of simultaneously embedding properly holomorphically into $\Bbb C^2$ a whole family of $n$-connected domains $Ω_r\subset\Bbb P^1$ such that none of the components of $\Bbb P^1\setminusΩ_r$ reduces to a point, by constructing a continuous mapping $Ξ\colon\bigcup_r\{r\}\timesΩ_r\to\Bbb C^2$ such that $Ξ(r,\cdot)\colonΩ_r\hookrightarrow\Bbb C^2$ is a proper holomorphic embedding for every $r$. To this aim, a parametric version of both the Andersén-Lempert procedure and Carleman's Theorem is formulated and proved.

math.CV↗
arXiv2021

Unipotent Factorization of Vector Bundle Automorphisms

We provide unipotent factorizations of vector bundle automorphisms of real and complex vector bundles over smooth manifolds. This generalises work of Thurston-Wasserstein and Wasserstein for trivial vector bundles. We also address two symplectic cases and propose a complex geometric analog of the problem in the setting of holomorphic vector bundles over Stein manifolds.

math.RA↗
arXiv2018

Runge tubes in Stein manifolds with the density property

In this paper we give a very simple proof of the existence and plenitude of Runge tubes in $\mathbb C^n$ $(n>1)$ and, more generally, in Stein manifolds with the density property. We show in particular that for any algebraic submanifold $A$ of codimension at least two in a complex Euclidean space $\mathbb C^n$, the normal bundle of $A$ in $\mathbb C^n$ admits a holomorphic embedding onto a Runge domain in $\mathbb C^n$ which agrees with the inclusion map $A\hookrightarrow \mathbb C^n$ on the zero section.

math.CV↗
arXiv2013

Polynomial Completion of Symplectic Jets and Surfaces Containing Involutive Lines

Motivated by work of Dragt and Abell on accelerator physics, we study the completion of symplectic jets by polynomial maps of low degrees. We use Andersén-Lempert Theory to prove that symplectic completions always exist, and we prove the degree bound conjectured by Dragt and Abell in the physically relevant cases. However, we disprove the degree bound for 3-jets in dimension 4. This follows from the fact that if $Σ$ is the disjoint union of $r=7$ involutive lines in $\mathbb P^3$, then $Σ$ is contained in a degree $d=4$ hypersurface, i.e., the restriction morphism $ι:H^0(\mathbb P^3,\mathcal O(4))\rightarrow H^0(Σ,\mathcal O(4))$ has a nontrivial kernel (Todd). We give two new proofs of this fact, and finally we show that if $(r,d)\neq (7,4)$ then the map $ι$ has maximal rank.

math.AG↗