Searcharxiv⌕ Search

arXiv subjects

Finnur Larusson

Publications and source records attributed to Finnur Larusson.

At least 19 recordsLinked to original sources
arXiv2026

Parametric equivariant Oka principle

Let $G$ be a reductive complex Lie group and $K$ be a maximal compact subgroup of $G$. Let $X$ be a reduced Stein $G$-space and $Y$ be a $G$-elliptic manifold. We prove the following parametric equivariant Oka principle. The inclusion of the space of holomorphic $G$-maps $X\to Y$ into the space of continuous $K$-maps $X\to Y$ is a weak homotopy equivalence with respect to the compact-open topology. The proof is divided into a homotopy-theoretic part, which is handled by an abstract theorem of Studer, and an analytic part, for which we prove equivariant versions of the homotopy approximation theorem and the nonlinear splitting lemma that are key tools in Oka theory. The principle can be strengthened so as to allow interpolation on a $G$-invariant subvariety of $X$ and approximation on a $K$-invariant holomorphically convex compact subset of $X$.

math.CV↗
arXiv2025

Generic conservative dynamics on Stein manifolds with the volume density property

We study the dynamics of generic volume-preserving automorphisms $f$ of a Stein manifold $X$ of dimension at least 2 with the volume density property. Among such $X$ are all connected linear algebraic groups (except $\mathbb{C}$ and $\mathbb{C}^*$) with a left- or right-invariant Haar form. We show that a generic $f$ is chaotic and of infinite topological entropy, and that the transverse homoclinic points of each of its saddle periodic points are dense in $X$. We present analogous results with similar proofs in the non-conservative case. We also prove the Kupka-Smale theorem in the conservative setting.

math.CV↗
arXiv2025

Dynamics of generic automorphisms of Stein manifolds with the density property

We study the dynamics of a generic automorphism $f$ of a Stein manifold with the density property. Such manifolds include all linear algebraic groups. Even in the special case of $\mathbb C^n$, $n\geq 2$, most of our results are new. We study the Julia set, non-wandering set, and chain-recurrent set of $f$. We show that the closure of the set of saddle periodic points of $f$ is the largest forward invariant set on which $f$ is chaotic. This subset of the Julia set of $f$ is also characterised as the closure of the set of transverse homoclinic points of $f$, and equals the Julia set if and only if a certain closing lemma holds. Among the other results in the paper is a generalisation of Buzzard's holomorphic Kupka-Smale theorem to our setting.

math.CV↗
arXiv2025

Regular immersions directed by algebraically elliptic cones

Let $M$ be an open Riemann surface and $A$ be the punctured cone in $\mathbb{C}^n\setminus\{0\}$ on a smooth projective variety $Y$ in $\mathbb{P}^{n-1}$. Recently, Runge approximation theorems with interpolation for holomorphic immersions $M\to\mathbb{C}^n$, directed by $A$, have been proved under the assumption that $A$ is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by $A$ from a smooth affine curve $M$ into $\mathbb{C}^n$. The Oka property is naturally replaced by the stronger assumption that $A$ is algebraically elliptic, which it is if $Y$ is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest.

math.CV↗
arXiv2024

Oka-1 manifolds: New examples and properties

In this paper we investigate Oka-1 manifolds and Oka-1 maps, a class of complex manifolds and holomorphic maps recently introduced by Alarcón and Forstnerič. Oka-1 manifolds are characterised by the property that holomorphic maps from any open Riemann surface to the manifold satisfy the Runge approximation and Weierstrass interpolation conditions, while Oka-1 maps enjoy similar properties for liftings of maps from open Riemann surfaces in the absence of topological obstructions. We also formulate and study the algebraic version of the Oka-1 condition, called aOka-1. We show that it is a birational invariant for compact algebraic manifolds and holds for all uniformly rational projective manifolds. This gives a Runge approximation theorem for maps from compact Riemann surfaces to uniformly rational projective manifolds. Finally, we study a class of complex manifolds with an approximation property for holomorphic sprays of discs. This class lies between the smaller class of Oka manifolds and the bigger class of Oka-1 manifolds and has interesting functorial properties.

math.CV↗
arXiv2024

A strong parametric h-principle for complete minimal surfaces

We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface $M$ into $\mathbb R^n$, $n\geq 3$. It follows that the inclusion of the space of such immersions into the space of all nonflat conformal minimal immersions is a weak homotopy equivalence. When $M$ is of finite topological type, the inclusion is a genuine homotopy equivalence. By a parametric h-principle due to Forstneric and Larusson, the space of complete nonflat conformal minimal immersions therefore has the same homotopy type as the space of continuous maps from $M$ to the punctured null quadric. Analogous results hold for holomorphic null curves $M\to\mathbb C^n$ and for full immersions in place of nonflat ones.

math.DG↗
arXiv2024

Isotopies of complete minimal surfaces of finite total curvature

Let $M$ be a Riemann surface biholomorphic to an affine algebraic curve. We show that the inclusion of the space $\Re \mathrm{NC}_*(M,\mathbb{C}^n)$ of real parts of nonflat proper algebraic null immersions $M\to\mathbb{C}^n$, $n\ge 3$, into the space $\mathrm{CMI}_*(M,\mathbb{R}^n)$ of complete nonflat conformal minimal immersions $M\to\mathbb{R}^n$ of finite total curvature is a weak homotopy equivalence. We also show that the $(1,0)$-differential $\partial$, mapping $\mathrm{CMI}_*(M,\mathbb{R}^n)$ or $\Re \mathrm{NC}_*(M,\mathbb{C}^n)$ to the space $\mathscr{A}^1(M,\mathbf{A})$ of algebraic $1$-forms on $M$ with values in the punctured null quadric $\mathbf{A} \subset \mathbb{C}^n\setminus\{0\}$, is a weak homotopy equivalence. Analogous results are obtained for proper algebraic immersions $M\to\mathbb{C}^n$, $n\ge 2$, directed by a flexible or algebraically elliptic punctured cone in $\mathbb{C}^n\setminus\{0\}$.

math.DG↗
arXiv2023

Gromov's Oka principle for equivariant maps

We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group $G$ acts on a Stein manifold $X$ and another manifold $Y$ in such a way that $Y$ is $G$-Oka, then every $G$-equivariant continuous map $X\to Y$ can be deformed, through such maps, to a $G$-equivariant holomorphic map. Approximation on a $G$-invariant holomorphically convex compact subset of $X$ and jet interpolation along a $G$-invariant subvariety of $X$ can be built into the theorem. We conjecture that the theorem holds for actions of arbitrary reductive complex Lie groups and prove partial results to this effect.

math.CV↗
arXiv2022

The space of Gauss maps of complete minimal surfaces

The Gauss map of a conformal minimal immersion of an open Riemann surface $M$ into $\mathbb R^3$ is a meromorphic function on $M$. In this paper, we prove that the Gauss map assignment, taking a full conformal minimal immersion $M\to\mathbb R^3$ to its Gauss map, is a Serre fibration. We then determine the homotopy type of the space of meromorphic functions on $M$ that are the Gauss map of a complete full conformal minimal immersion, and show that it is the same as the homotopy type of the space of all continuous maps from $M$ to the 2-sphere. We obtain analogous results for the generalised Gauss map of conformal minimal immersions $M\to\mathbb R^n$ for arbitrary $n\geq 3$.

math.DG↗
arXiv2022

Holomorphic Legendrian curves in projectivised cotangent bundles

We study holomorphic Legendrian curves in the standard complex contact structure on the projectivised cotangent bundle $X=\mathbb P(T^*Z)$ of a complex manifold $Z$ of dimension at least $2$. We provide a detailed analysis of Legendrian curves degenerating to vertical curves and obtain several approximation and general position theorems. In particular, we prove that any vertical holomorphic curve $M\to X$ from a compact bordered Riemann surface $M$ can be deformed to a horizontal Legendrian curve by an arbitrarily small deformation. A similar result is proved in the parametric setting, provided that all vertical curves under consideration are nondegenerate. Stronger results are obtained when the base $Z$ is an Oka manifold or a Stein manifold with the density property. Finally, we establish basic and 1-parametric h-principles for holomorphic Legendrian curves in $X$.

math.CV↗
arXiv2022

Holomorphic Legendrian curves in $\mathbb{CP}^3$ and superminimal surfaces in $\mathbb S^4$

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into $\mathbb{CP}^3$ as a complete holomorphic Legendrian curve. Under the twistor projection $π:\mathbb{CP}^3\to \mathbb S^4$ onto the $4$-sphere, immersed holomorphic Legendrian curves $M\to \mathbb{CP}^3$ are in bijective correspondence with superminimal immersions $M\to\mathbb S^4$ of positive spin according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in $\mathbb S^4$. In particular, superminimal immersions into $\mathbb S^4$ satisfy the Runge approximation theorem and the Calabi-Yau property.

math.DG↗
arXiv2021

Equivariant Oka theory: Survey of recent progress

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group $G$ acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle $E$ of homogeneous spaces for a group bundle $\mathscr G$, all over a reduced Stein space $X$ with compatible actions of a reductive complex group on $E$, $\mathscr G$, and $X$. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov's Oka principle based on a new notion of a $G$-manifold being $G$-Oka.

math.CV↗
arXiv2021

Dynamics of generic endomorphisms of Oka-Stein manifolds

We study the dynamics of a generic endomorphism $f$ of an Oka-Stein manifold $X$. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several descriptions of the Fatou set and the Julia set of $f$. In particular, we show that the Julia set is the derived set of the set of attracting periodic points of $f$ and that it is also the closure of the set of repelling periodic points of $f$. Among other results, we prove that $f$ is chaotic on the Julia set and that every periodic point of $f$ is hyperbolic. We also give an explicit description of the "Conley decomposition" of $X$ induced by $f$ into chain-recurrence classes and basins of attractors. For $X=\mathbb{C}$, we prove that every Fatou component is a disc and that every point in the Fatou set is attracted to an attracting cycle or lies in a dynamically bounded wandering domain (whether such domains exist is an open question).

math.CV↗
arXiv2019

Generic aspects of holomorphic dynamics on highly flexible complex manifolds

We prove closing lemmas for automorphisms of a Stein manifold with the density property and for endomorphisms of an Oka-Stein manifold. In the former case we need to impose a new tameness condition. It follows that hyperbolic periodic points are dense in the tame non-wandering set of a generic automorphism of a Stein manifold with the density property and in the non-wandering set of a generic endomorphism of an Oka-Stein manifold. These are the first results about holomorphic dynamics on Oka manifolds. We strengthen previous results of ours on the existence and genericity of chaotic volume-preserving automorphisms of Stein manifolds with the volume density property. We build on work of Fornaess and Sibony: our main results generalise theorems of theirs and we use their methods of proof.

math.CV↗
arXiv2019

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic analogues of three of the central properties of analytic Oka theory fail for all compact manifolds and manifolds with a rational curve; in particular, for projective manifolds.

math.AG↗
arXiv2018

Chaotic holomorphic automorphisms of Stein manifolds with the volume density property

Let $X$ be a Stein manifold of dimension $n\geq 2$ satisfying the volume density property with respect to an exact holomorphic volume form. For example, $X$ could be $\mathbb{C}^n$, any connected linear algebraic group that is not reductive, the Koras-Russell cubic, or a product $Y\times\mathbb{C}$, where $Y$ is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of $X$. In particular, $X$ has a chaotic holomorphic automorphism. A proof for $X=\mathbb{C}^n$ may be found in work of Fornæss and Sibony. We follow their approach closely. Peters, Vivas, and Wold showed that a generic volume-preserving automorphism of $\mathbb{C}^n$, $n\geq 2$, has a hyperbolic fixed point whose stable manifold is dense in $\mathbb{C}^n$. This property can be interpreted as a kind of chaos. We generalise their theorem to a Stein manifold as above.

math.CV↗
arXiv2017

The Oka principle for holomorphic Legendrian curves in $\mathbb C^{2n+1}$

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ of continuous maps from $M$ to the sphere $\mathbb S^{4n-1}$. If $M$ has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of $\mathscr L(M,\mathbb C^{2n+1})$ in terms of the homotopy groups of $\mathbb S^{4n-1}$. It follows that $\mathscr L(M,\mathbb C^{2n+1})$ is $(4n-3)$-connected.

math.DG↗