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Erlend Fornaess Wold

Publications and source records attributed to Erlend Fornaess Wold.

At least 19 recordsLinked to original sources
arXiv2022

Oka domains in Euclidean spaces

In this paper we find surprisingly small Oka domains in Euclidean spaces $\mathbb C^n$ of dimension $n>1$ at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set $E$ in $\mathbb C^n$ we show that $\mathbb C^n\setminus E$ is an Oka domain. In particular, there are Oka domains which are only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives families of smooth real hypersurfaces $Σ_t\subset \mathbb C^n$ $(t\in\mathbb R)$ dividing $\mathbb C^n$ in an unbounded hyperbolic domain and an Oka domain such that at the threshold value $t=0$ the hypersurface $Σ_0$ is a hyperplane and the character of the two sides gets reversed. More generally, we show that if $E$ is a closed set in $\mathbb C^n$ for $n>1$ whose projective closure $\overline E\subset\mathbb C\mathbb P^n$ avoids a hyperplane $Λ\subset\mathbb C\mathbb P^n$ and is polynomially convex in $\mathbb C\mathbb P^n\setminus Λ\cong\mathbb C^n$, then $\mathbb C^n\setminus E$ is an Oka domain.

math.CV↗
arXiv2021

Holomorphic families of Fatou-Bieberbach domains and applications to Oka manifolds

We construct holomorphically varying families of Fatou-Bieberbach domains with given centres in the complement of any compact polynomially convex subset $K$ of $\mathbb C^n$ for $n>1$. This provides a simple proof of the recent result of Yuta Kusakabe to the effect that the complement $\mathbb C^n\setminus K$ of any polynomially convex subset $K$ of $\mathbb C^n$ is an Oka manifold. The analogous result is obtained with $\mathbb C^n$ replaced by any Stein manifold with the density property.

math.CV↗
arXiv2020

Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-Weil, and Mergelyan

In this paper we survey the theory of holomorphic approximation, from the classical 19th century results of Runge and Weierstrass, continuing with the 20th century work of Oka and Weil, Mergelyan, Vitushkin and others, to the most recent ones on higher dimensional manifolds. The paper includes some new results and applications of this theory, especially to manifold-valued maps.

math.CV↗
arXiv2018

Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains

We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the infinitesimal Carathéodory metric $g_C(z,v)$ and the infinitesimal Kobayashi metric $g_K(z,v)$ coincide if $z$ is sufficiently close to $bD$ and if $v$ is sufficiently close to being tangential to $bD$. Also, we show that every two close points of $D$ sufficiently close to the boundary and whose difference is almost tangential to $bD$ can be joined by a (unique up to reparameterization) complex geodesic of $D$ which is also a holomorphic retract of $D$. The same continues to hold if $D$ is a worm domain, as long as the points are sufficiently close to a strongly pseudoconvex boundary point. We also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed, this has consequences for the behavior of the squeezing function.

math.CV↗
arXiv2017

Totally real embeddings with prescribed polynomial hulls

We embed compact $C^\infty$ manifolds into $\mathbb C^n$ as totally real manifolds with prescribed polynomial hulls. As a consequence we show that any compact $C^\infty$ manifold of dimension $d$ admits a totally real embedding into $\mathbb C^{\lfloor \frac{3d}{2}\rfloor}$ with non-trivial polynomial hull without complex structure.

math.CV↗
arXiv2016

Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces

We prove that all locally exposable points in a Stein compact in a complex space can be exposed along a given curve to a given real hypersurface. Moreover, the exposing map for a boundary point can be sufficiently close to the identity map outside any fixed neighborhood of the point. We also prove a parametric version of this result for bounded strongly pseudoconvex domains in $\mathbb C^n$. For a bounded strongly pseudoconvex domain in $\mathbb C^n$ and a given boundary point of it, we prove that there is a global coordinate change on the closure of the domain which is arbitrarily close to the identity map with respect to the $C^1$-norm and maps the boundary point to a strongly convex boundary point.

math.CV↗
arXiv2014

Fatou-Bieberbach domains in $\mathbb C^n\setminus\mathbb R^k$

We construct Fatou-Bieberbach domains in $\mathbb C^n$ for $n>1$ which contain a given compact set $K$ and at the same time avoid a totally real affine subspace $L$ of dimension $<n$, provided that $K\cup L$ is polynomially convex. By using this result, we show that the domain $\mathbb C^n\setminus\mathbb R^k$ for $1\le k<n$ enjoys the Oka property with approximation for maps from any Stein manifold of dimension $<n$.

math.CV↗
arXiv2014

Proper holomorphic embeddings into Stein manifolds with the density property

We prove that a Stein manifold of dimension $d$ admits a proper holomorphic embedding into any Stein manifold of dimension at least $2d+1$ satisfying the holomorphic density property. This generalizes classical theorems of Remmert, Bishop and Narasimhan pertaining to embeddings into complex Euclidean spaces, as well as several other recent results.

math.CV↗
arXiv2013

Free dense subgroups of holomorphic automorphisms

We show the existence of free dense subgroups, generated by 2 elements, in the holomorphic shear and overshear group of complex-Euklidean space and extend this result to the group of holomorphic automorphisms of Stein manifolds with Density Property, provided there exists a generalized translation. The conjugation operator associated to this generalized translation is hypercyclic on the topological space of holomorphic automorphisms.

math.CV↗