Searcharxiv⌕ Search

arXiv subjects

Franc Forstneric

Publications and source records attributed to Franc Forstneric.

At least 37 records · Page 2Linked to original sources
arXiv2023

Hyperbolic domains in real Euclidean spaces

The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb R^n$, $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $\mathbb R^n$, the minimal metric coincides with the classical Beltrami-Cayley-Klein metric, one of the models of hyperbolic geometry. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric.

math.CV↗
arXiv2023

Proper holomorphic embeddings with small limit sets

Let $X$ be a Stein manifold of dimension $n\ge 1$. Given a continuous positive increasing function $h$ on $\mathbb R_+=[0,\infty)$ with $\lim_{t\to\infty} h(t)=\infty$, we construct a proper holomorphic embedding $f=(z,w):X\hookrightarrow \mathbb C^{n+1}\times \mathbb C^n$ satisfying $|w(x)|<h(|z(x)|)$ for all $x\in X$. In particular, $f$ may be chosen such that its limit set at infinity is a linearly embedded copy of $\mathbb{CP}^n$ in $\mathbb{CP}^{2n}$.

math.CV↗
arXiv2023

Domains without parabolic minimal submanifolds and weakly hyperbolic domains

We show that if $Ω$ is an $m$-convex domain in $\mathbb R^n$ for some $2\le m<n$ whose boundary $bΩ$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $Ω$ does not contain any immersed parabolic minimal submanifold of dimension $\ge m$. In particular, if $M$ is a properly embedded nonflat minimal hypersurface in $\mathbb R^n$ with a tubular neighbourhood of positive radius then every immersed parabolic hypersurface in $\mathbb R^n$ intersects $M$. In dimension $n=3$ this holds if $M$ has bounded Gaussian curvature. We also introduce the class of weakly hyperbolic domains $Ω$ in $\mathbb R^n$ characterised by the property that every conformal harmonic map $\mathbb C\toΩ$ is constant, and we elucidate their relationship with hyperbolic domains and domains without parabolic minimal surfaces.

math.DG↗
arXiv2023

Embedded complex curves in the affine plane

This paper brings several contributions to the classical Forster-Bell-Narasimhan conjecture and the Yang problem concerning the existence of proper and almost proper (hence complete) injective holomorphic immersions of open Riemann surfaces in the affine plane $\mathbb C^2$ satisfying interpolation and hitting conditions. We also show that in every compact Riemann surface there is a Cantor set whose complement admits a proper holomorphic embedding in $\mathbb C^2$. The focal point is a lemma saying the following. Given a compact bordered Riemann surface, $M$, a closed discrete subset $E$ of its interior $\mathring M=M\setminus bM$, a compact subset $K\subset \mathring M\setminus E$ without holes in $\mathring M$, and a $\mathscr C^1$ embedding $f:M\hookrightarrow \mathbb C^2$ which is holomorphic in $\mathring M$, we can approximate $f$ uniformly on $K$ by a holomorphic embedding $F:M\hookrightarrow \mathbb C^2$ which maps $E\cup bM$ out of a given ball and satisfies some interpolation conditions.

math.CV↗
arXiv2023

Recent developments on Oka manifolds

In this paper we present the main developments in Oka theory since the publication of my book Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis)}, Second Edition, Springer, 2017. We also give several new results, examples and constructions of Oka domains in Euclidean and projective spaces. Furthermore, we show that for $n>1$ the fibre $\mathbb C^n$ in a Stein family can degenerate to a non-Oka fibre, thereby answering a question of Takeo Ohsawa. Several open problems are discussed.

math.CV↗
arXiv2023

Embedding bordered Riemann surfaces in strongly pseudoconvex domains

We show that every bordered Riemann surface, $M$, with smooth boundary $bM$ admits a proper holomorphic map $M\to Ω$ into any bounded strongly pseudoconvex domain $Ω$ in $\mathbb C^n$, $n>1$, extending to a smooth map $f:\overline M\to\overline Ω$ which can be chosen an immersion if $n\ge 3$ and an embedding if $n\ge 4$. Furthermore, $f$ can be chosen to approximate a given holomorphic map $\overline M\to Ω$ on compacts in $M$ and interpolate it at finitely many given points in $M$.

math.CV↗
arXiv2023

Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets

We show that if $E$ is a closed convex set in $\mathbb C^n$ $(n>1)$ contained in a closed halfspace $H$ such that $E\cap bH$ is nonempty and bounded, then the concave domain $Ω= \mathbb C^n\setminus E$ contains images of proper holomorphic maps $f:X\to \mathbb C^n$ from any Stein manifold $X$ of dimension $<n$, with approximation of a given map on closed compact subsets of $X$. If in addition $2\dim X+1\le n$ then $f$ can be chosen an embedding, and if $2\dim X=n$ then it can be chosen an immersion. Under a stronger condition on $E$ we also obtain the interpolation property for such maps on closed complex subvarieties.

math.CV↗
arXiv2022

Flexible domains for minimal surfaces in Euclidean spaces

In this paper we introduce and investigate a new notion of flexibility for domains in Euclidean spaces $\mathbb R^n$ for $n\ge 3$ in terms of minimal surfaces which they contain. A domain $Ω$ in $\mathbb R^n$ is said to be flexible if every conformal minimal immersion $U\toΩ$ from a Runge domain $U$ in an open conformal surface $M$ can be approximated uniformly on compacts, with interpolation on any given finite set, by conformal minimal immersion $M\to Ω$. Together with hyperbolicity phenomena considered in recent works, this extends the dichotomy between flexibility and rigidity from complex analysis to minimal surface theory.

math.DG↗
arXiv2022

The Calabi-Yau problem for minimal surfaces with Cantor ends

We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in $\mathbb R^3$ with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least $2$, for holomorphic null immersions into $\mathbb C^n$ with $n\ge 3$, for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions in any self-dual or anti-self-dual Einstein four-manifold.

math.DG↗
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↗
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↗
arXiv2022

Every smoothly bounded p-convex domain in R^n admits a p-plurisubharmonic defining function

We show that every bounded domain $D$ in $\mathbb R^n$ with smooth $p$-convex boundary for $2\le p < n$ admits a smooth defining function $ρ$ which is $p$-plurisubharmonic on $\overline D$; if in addition $bD$ has no $p$-flat points then $ρ$ can be chosen strongly $p$-plurisubharmonic on $D$. If $bD$ is $2$-convex then for any open connected conformal surface $M$ and conformal harmonic map $f:M\to \overline D$, either $f(M)\subset D$ or $f(M)\subset bD$. In particular, every conformal harmonic map $\mathbb D^*\to D$ from the punctured disc extends to a conformal harmonic map $\mathbb D\to D$.

math.CV↗
arXiv2021

The first thirty years of Andersen-Lempert theory

In this paper we expose the impact of the fundamental discovery, made by Erik Andersén and László Lempert in 1992, that the group generated by shears is dense in the group of holomorphic automorphisms of complex Euclidean spaces of dimensions $n>1$. In three decades since its publication, their groundbreaking work led to the discovery of several new phenomena and to major new results in complex analysis and geometry involving Stein manifolds and affine algebraic manifolds with many automorphisms. The aim of this survey is to present some focal points of these developments, with a view towards the future.

math.CV↗
arXiv2021

Schwarz-Pick lemma for harmonic maps which are conformal at a point

We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $\mathbb D$ in $\mathbb C$ into the unit ball $\mathbb B^n$ in $\mathbb R^n$, $n\ge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $n\ge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $\mathbb D\to \mathbb B^n$. This implies that conformal harmonic immersions $M \to \mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$\mathrm{é}$ metric on $M$ and the Cayley-Klein metric on the ball $\mathbb B^n$, and the extremal maps are precisely the conformal embeddings of the disc $\mathbb D$ onto affine discs in $\mathbb B^n$. By using these results, we lay the foundations of the hyperbolicity theory for domains in $\mathbb R^n$ based on minimal surfaces.

math.DG↗
arXiv2021

Euclidean domains in complex manifolds

In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets of the form $K\cup M$ in a complex manifold $X$, where $K$ is a compact $\mathscr O(U)$-convex set in an open Stein neighbourhood $U$ of $K$, $M$ is an embedded Stein submanifold of $X$, and $K\cap M$ is compact and $\mathscr O(M)$-convex. We prove a Docquier-Grauert type theorem concerning biholomorphic equivalence of neighbourhoods of such sets, and we give sufficient conditions for the existence of Stein neighbourhoods of $K\cup M$, biholomorphic to domains in $\mathbb C^n$ with $n=\dim X$, such that $M$ is mapped onto a closed complex submanifold of $\mathbb C^n$.

math.CV↗