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Erlend Fornæss Wold

Publications and source records attributed to Erlend Fornæss Wold.

16 recordsLinked to original sources
arXiv2026

Holomorphic Approximation for Real Diffeomorphism Groups

We show that every diffeomorphism of $\mathbb{R}^n$ for $n \ge 2$ can be approximated by an automorphism of $\mathbb{C}^n$ in the Whitney $C^k$-topology for any positive integer $k$ using the notion of density property. More precisely we find sufficient conditions for this holomorphic approximation of diffeomorphisms to hold and prove that the split real forms of most linear algebraic groups satisfy these conditions. In the same manner we also show holomorphic approximations of volume-preserving diffeomorphisms for the split real forms of linear algebraic groups equipped with the left-invariant volume form.

math.CV↗
arXiv2019

Hamiltonian Carleman approximation and the density property for coadjoint orbits

For a complex Lie group $G$ with a real form $G_0\subset G$, we prove that any Hamiltionian automorphism $ϕ$ of a coadjoint orbit $\mathcal O_0$ of $G_0$ whose connected components are simply connected, may be approximated by holomorphic $\mathcal O_0$-invariant symplectic automorphism of the corresponding coadjoint orbit of $G$ in the sense of Carleman, provided that $\mathcal O$ is closed. In the course of the proof, we establish the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.

math.SG↗
arXiv2017

Squeezing functions and Cantor Sets

We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function. Finally we show that complements of Cantor sets arising as Julia sets of quadratic polynomials have degenerate squeezing functions, despite of having Hausdorff dimension arbitrarily close to two.

math.CV↗
arXiv2017

Asymptotics of Invariant Metrics in the normal direction and a new characterisation of the unit disk

We give improvements of estimates of invariant metrics in the normal direction on strictly pseudoconvex domains. Specifically we will give the second term in the expansion of the metrics. This depends on an improved localisation result and estimates in the one variable case. Finally we will give a new characterisation of the unit disk in $\mathbb C$ in terms of the asymptotic behaviour of quotients of invariant metrics.

math.CV↗
arXiv2016

A characterization of the ball

We study bounded domains with certain smoothness conditions and the properties of their squeezing functions in order to prove that the domains are biholomorphic to the ball.

math.CV↗
arXiv2014

Presence or absence of analytic structure in maximal ideal spaces

We study extensions of Wermer's maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold.

math.CV↗
arXiv2013

Embedding univalent functions in filtering Loewner chains in higher dimension

We discuss the problem of embedding univalent functions into Loewner chains in higher dimension. In particular, we prove that a normalized univalent map of the ball in $\C^n$ whose image is a smooth strongly pseudoconvex domain is embeddable into a normalized Loewner chain (satisfying also some extra regularity properties) if and only if the closure of the image is polynomially convex.

math.CV↗
arXiv2011

Riemann surfaces in Stein manifolds with density property

It is shown that any open Riemann surface can be immersed in any Stein manifold with (volume) density property and of dimension at least 2, if the manifold possesses an exhaustion with holomorphically convex compacts such that their complement is connected. The immersion can be made into an embedding if the dimension is at least 3. As an application, it is shown that Stein manifolds with (volume) density property and of dimension at least 3, are characterized among all other complex manifolds by their semigroup of holomorphic endomorphisms.

math.CV↗
arXiv2011

Uniform algebras and approximation on manifolds

Let $Ω\subset \mathbb{C}^n$ be a bounded domain and let $\mathcal{A} \subset \mathcal{C}(\barΩ)$ be a uniform algebra generated by a set $F$ of holomorphic and pluriharmonic functions. Under natural assumptions on $Ω$ and $F$ we show that the only obstruction to $\mathcal{A} = \mathcal{C}(\barΩ)$ is that there is a holomorphic disk $D \subset \barΩ$ such that all functions in $F$ are holomorphic on $D$, i.e., the only obstruction is the obvious one. This generalizes work by A. Izzo. We also have a generalization of Wermer's maximality theorem to the (distinguished boundary of the) bidisk.

math.CV↗
arXiv2006

Attracting basins of volume preserving automorphisms of $\mathbb{C}^k$

We study toplogical properties of attracting sets for automorphisms of $\mathbb{C}^k$. Our main result is that a generic volume preserving automorphism has a hyperbolic fixed point with a dense stable manifold. We prove the same result for volume preserving maps tangent to the identity. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if the fixed point is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting.

math.CV↗
arXiv2005

Fatou-Bieberbach Domains

We show that for any $m\in\NN\cup\{\infty\}$ there exist $m$ disjoint FB domains whose union is dense in $\CC^k$. In fact we show that any point not in the union is a boundary point for all the domains. We construct FB domains that contains arbitrary countable collections of subvarieties of $\CC^k$, and we construct FB domains that intersect elements of countable collections of affine subspaces of $\CC^k$ in connected proper subsets. Moreover, we show that any Runge FB domain is the attracting basin for a sequence of automorphisms of $\CC^k$, although not necessarily if you only allow iteration of one automorphism. We also show that an increasing sequence of Runge $\CC^k$'s is a $\CC^k$.

math.CV↗
arXiv2004

Non-Autonomous Basins of Attraction With 4-Dimensional Boundaries

We study whether the basin of attraction of a sequence of automorphisms of $\mathbb{C}^k$ is biholomorphic to $\mathbb{C}^k$. In particular we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to $\mathbb{C}^k$ if the maps are repeated often enough. We also construct Fatou-Bieberbach domains whose boundaries are 4-dimensional.

math.CV↗