arXiv subjects

Franc Forstneric

Publications and source records attributed to Franc Forstneric.

At least 19 recordsLinked to original sources

arXiv2026

Families of proper minimal surfaces

Assume that $X$ is a connected, open, oriented smooth surface, $B$ is a compact Euclidean neighbourhood retract, and $\mathscr{J}=\{J_b\}_{b\in B}$ is a continuous family of complex structures on $X$ of local H\"older class $\mathscr{C}^\alpha$ for some $0<\alpha<1$. We construct a continuous family of $J_b$-conformal minimal immersions $u_b:X\to \mathbb{R}^3$, $b\in B$, properly projecting to $\mathbb{R}^2$ and having an arbitrary given family of flux homomorphisms ${\rm Flux}_{u_b}:H_1(X,\mathbb{Z})\to\mathbb{R}^3$. In particular, there are continuous families of proper $J_b$-holomorphic null immersions $X\to \mathbb{C}^3$ and of proper $J_b$-holomorphic immersions $X\to\mathbb{C}^2$, $b\in B$.

math.DG↗

arXiv2026

On an extremal problem for harmonic maps conformal at a point

Let $\mathbb D$ denote the unit disc in $\mathbb C$. For a domain $D\subset\mathbb C$ and a point $p\in D$, let $M_D(p)$ denote the supremum of $\|df_0\|$ over all harmonic maps $f:\mathbb D\to D$ with $f(0)=p$ whose differential $df_0$ at $0\in \mathbb D$ is conformal. If $f:\mathbb D\to D$ is a conformal diffeomorphism onto $D$ with $f(0)=p$, then $\|df_0\|\le M_D(p)$. In a recent paper, the authors proved that equality holds when $D=\mathbb D$, and they asked whether equality can hold only when $D$ is a round disc. We give a negative answer by proving that, among bounded convex pointed domains $p\in D\subset\mathbb C$ and up to translations, rotations, and reflections, equality holds if and only if, after moving $p$ to the origin, $D=F(\mathbb D)$ where $F:\mathbb D\to\mathbb C$ is a holomorphic map with $F(0)=0$ and $F'(z)=\frac{c}{1+az+\lambda z^2}$, where $c>0$, $|\lambda|<1$, and $|a-\bar a\lambda|<1-|\lambda|^2$. This family contains strongly convex examples which are not round discs.

math.CV↗

arXiv2026

The Oka principle for holomorphic fibre bundles of Holder-Zygmund classes on strongly pseudoconvex domains

Let $\overline \Omega$ be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let $h:Z\to \overline \Omega$ be a fibre bundle of H\"older-Zygmund class $\Lambda^r$, $r>0$, which is holomorphic over $\Omega$. Assuming that the fibre is an Oka manifold, we prove that every continuous section $f_0:\overline \Omega\to Z$ is homotopic to a section $f_1:\overline \Omega\to Z$ of class $\Lambda^r(\overline \Omega)$ which is holomorphic on $\Omega$. We also establish the parametric h-principle in this context. As an application, we obtain the Oka principle for the classification of vector bundles and principal bundles of H\"older-Zygmund classes on such domains.

math.CV↗

arXiv2025

Removing singularities of minimal surfaces by isotopies

Given an open Riemann surface $M$, we show that the branch points and the complete ends of finite total curvature of a conformal minimal surface $M\to{\mathbb R}^n$, $n\ge 3$, can be removed by an isotopy through such surfaces. The analogous result holds for null holomorphic curves $M\to{\mathbb C}^n$.

math.DG↗

arXiv2025

The universal family of punctured Riemann surfaces is Stein

We show that the universal Teichm\"uller family of n-punctured compact Riemann surfaces of genus g is a Stein manifold for any n>0. We describe its basic function theoretic properties and pose several challenging questions. We show in particular that the space of fibrewise algebraic functions on the universal family is dense in the space of holomorphic functions, and there is a fibrewise algebraic map of the universal family in a Euclidean space which restricts to a proper embedding on every fibre. We also obtain a relative Oka principle for holomorphic fibrewise algebraic maps of the universal family to any flexible algebraic manifold.

math.CV↗

arXiv2025

The Oka principle for tame families of Stein manifolds

Let $X$ be a smooth open manifold of even dimension, $T$ be a topological space, and $\mathscr{J}=\{J_t\}_{t\in T}$ be a continuous family of smooth integrable Stein structures on $X$. Under suitable additional assumptions on $T$ and $\mathscr{J}$, we prove an Oka principle for continuous families of maps from the family of Stein manifolds $(X,J_t)$, $t\in T$, to any Oka manifold, showing that every family of continuous maps is homotopic to a family of $J_t$-holomorphic maps depending continuously on $t$. We also prove the Oka-Weil theorem for sections of $\mathscr{J}$-holomorphic vector bundles on $Z=T\times X$ and the Oka principle for isomorphism classes of such bundles. The assumption on the family $\mathscr{J}$ is that the $J_t$-convex hulls of any compact set in $X$ are upper semicontinuous with respect to $t\in T$; such a family is said to be tame. For suitable parameter spaces $T$, we characterise tameness by the existence of a continuous family $\rho_t:X\to \mathbb{R}_+=[0,+\infty)$, $t\in T$, of strongly $J_t$-plurisubharmonic exhaustion functions on $X$. Every family of complex structures on an open orientable surface is tame.We give an example of a nontame smooth family of Stein structures $J_t$ on $\R^{2n}$ $(t\in \mathbb{R},\ n>1)$ such that $(\mathbb{R}^{2n},J_t)$ is biholomorphic to $\mathbb{C}^n$ for every $t\in\mathbb{R}$. We show that the Oka principle fails on any nontame family.

math.CV↗

arXiv2025

From Stein manifolds to Oka manifolds: the h-principle in complex analysis

This introduction to the homotopy principle in complex analysis and geometry, better known as the Oka theory, is aimed at wide mathematical audience. After a brief historical survey of the h-principle in smooth analysis and geometry, I present the key notions of Oka manifolds and Oka maps, which developed from the Oka-Grauert principle and Gromov's theory of elliptic complex manifolds and elliptic holomorphic submersions. I discuss recent and ongoing developments, open problems, and mention some applications. The paper also includes a brief survey of the recently developed h-principles in the classical theory of minimal surfaces.

math.CV↗

arXiv2025

The nonhomogeneous Cauchy-Riemann equation on families of open Riemann surfaces

In this paper we use the nonhomogeneous Beltrami equation to give an optimal solution to the nonhomogeneous Cauchy-Riemann equation for continuous or smooth families of complex structures and $(0,1)$-forms of a H\"older class on a smooth open orientable surface. As an application, we obtain the Oka-Grauert principle for complex line bundles on families of open Riemann surfaces.

math.CV↗

arXiv2025

Approximation of biholomorphic maps between Runge domains by holomorphic automorphisms

We show that biholomorphic maps between certain pairs of Runge domains in the complex affine space $\mathbb C^n$, $n>1$, are limits of holomorphic automorphisms of $\mathbb C^n$. A similar result holds for volume preserving maps and also in Stein manifolds with the density property. This generalizes several results in the literature and provides a considerably simpler proof.

math.CV↗

arXiv2025

Every projective Oka manifold is elliptic

We show that every projective Oka manifold is elliptic in the sense of Gromov. This gives an affirmative answer to a long-standing open question.

math.CV↗

arXiv2024

Runge and Mergelyan theorems on families of open Riemann surfaces

Given a smooth open oriented surface $X$, endowed with a family of complex structures $\{J_b\}_{b\in B}$ of some H\"older class and depending continuously or smoothly on the parameter $b$ in a suitable topological space $B$, we construct continuous or smooth families $F_b:X\to Y$, $b\in B$, of $J_b$-holomorphic maps to any Oka manifold $Y$, with approximation on a suitable family of compact Runge sets in $X$. Along the way, we prove Runge and Mergelyan approximation theorems and Weierstrass interpolation theorem for functions on such families. We include applications to the construction of families of directed holomorphic immersions and conformal minimal immersions to Euclidean spaces.

math.CV↗

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↗

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 Alarc\'on and Forstneri\v{c}. 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↗

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

Oka tubes in holomorphic line bundles

Let $(E,h)$ be a semipositive hermitian holomorphic line bundle on a compact complex manifold $X$ with $\dim X>1$. Assume that for each point $x\in X$ there exists a divisor $D\in |E|$ in the complete linear system determined by $E$ whose complement $X\setminus D$ is a Stein neighbourhood of $x$ with the density property. Then, the disc bundle $\Delta_h(E)=\{e\in E:|e|_h<1\}$ is an Oka manifold while $D_h(E)=\{e\in E:|e|_h>1\}$ is a Kobayashi hyperbolic domain. In particular, the zero section of $E$ admits a basis of Oka neighbourhoods $\{|e|_h 0$. We show that this holds if $X$ is a rational homogeneous manifold of dimension $>1$. This class of manifolds includes complex projective spaces, Grassmannians, and flag manifolds. This phenomenon contributes to the heuristic principle that Oka properties are related to metric positivity of complex manifolds.

math.CV↗

arXiv2023

Minimal surfaces with symmetries

Let $G$ be a finite group acting on a connected open Riemann surface $X$ by holomorphic automorphisms and acting on a Euclidean space $\mathbb R^n$ $(n\ge 3)$ by orthogonal transformations. We identify a necessary and sufficient condition for the existence of a $G$-equivariant conformal minimal immersion $F:X\to\mathbb R^n$. We show in particular that such a map $F$ always exists if $G$ acts without fixed points on $X$. Furthermore, every finite group $G$ arises in this way for some open Riemann surface $X$ and $n=2|G|$. We obtain an analogous result for minimal surfaces having complete ends with finite total Gaussian curvature, and for discrete infinite groups acting on $X$ properly discontinuously and acting on $\mathbb R^n$ by rigid transformations.

math.DG↗

arXiv2023

Complete nonsingular holomorphic foliations on Stein manifolds

Let $X$ be a Stein manifold of complex dimension $n>1$ endowed with a Riemannian metric $\mathfrak{g}$. We show that for every integer $k$ with $\left[\frac{n}{2}\right] \le k \le n-1$ there is a nonsingular holomorphic foliation of dimension $k$ on $X$ all of whose leaves are topologically closed and $\mathfrak{g}$-complete. The same is true if $1\le k<\left[\frac{n}{2}\right]$ provided that there is a complex vector bundle epimorphism $TX\to X\times\mathbb{C}^{n-k}$. We also show that if $\mathcal{F}$ is a proper holomorphic foliation on $\mathbb{C}^n$ $(n>1)$ then for any Riemannian metric $\mathfrak{g}$ on $\mathbb{C}^n$ there is a holomorphic automorphism $\Phi$ of $\mathbb{C}^n$ such that the image foliation $\Phi_*\mathcal{F}$ is $\mathfrak{g}$-complete. The analogous result is obtained on every Stein manifold with Varolin's density property.

math.CV↗

arXiv2023

Oka-1 manifolds

In this paper we begin a systematic study of the class of complex manifolds which are universal targets of holomorphic maps from open Riemann surfaces. We call them Oka-1 manifolds, by analogy with Oka manifolds that are universal targets of holomorphic maps from Stein manifolds of arbitrary dimension. We prove that every complex manifold which is dominable at most points by spanning tubes of complex lines in affine spaces is an Oka-1 manifold. In particular, a manifold dominable by $\mathbb{C}^n$ at most points is an Oka-1 manifold. We provide many examples of Oka-1 manifolds among compact complex surfaces, including all Kummer surfaces and all elliptic K3 surfaces. We show that the class of Oka-1 manifolds is invariant under Oka-1 maps inducing a surjective homomorphism of fundamental groups; this includes holomorphic fibre bundles with connected Oka fibres. In another direction, we prove that every bordered Riemann surface admits a holomorphic map with dense image in any connected complex manifold. The analogous result is shown for holomorphic Legendrian immersions in an arbitrary connected complex contact manifold.

math.CV↗