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Franc Forstneric

Publications and source records attributed to Franc Forstneric.

At least 73 records · Page 4Linked to original sources
arXiv2018

New complex analytic methods in the theory of minimal surfaces: a survey

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann-Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi-Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^n$ and in minimally convex domains of $\mathbb{R}^n$, results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

math.DG↗
arXiv2017

Every meromorphic function is the Gauss map of a conformal minimal surface

Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a holomorphic null curve $M\to\mathbb{C}^3$. Analogous results are proved for conformal minimal immersions $M\to\mathbb{R}^n$ for any $n>3$. We also show that every conformal minimal immersion $M\to\mathbb{R}^n$ is isotopic through conformal minimal immersions $M\to\mathbb{R}^n$ to a flat one, and we identify the path connected components of the space of all conformal minimal immersions $M\to\mathbb{R}^n$ for any $n\ge 3$.

math.DG↗
arXiv2017

Complete densely embedded complex lines in $\mathbb{C}^2$

In this paper we construct a complete injective holomorphic immersion $\mathbb{C}\to\mathbb{C}^2$ whose image is dense in $\mathbb{C}^2$. The analogous result is obtained for any closed complex submanifold $X\subset \mathbb{C}^n$ for $n>1$ in place of $\mathbb{C}\subset\mathbb{C}^2$. We also show that, if $X$ intersects the unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ and $K$ is a connected compact subset of $X\cap\mathbb{B}^n$, then there is a Runge domain $Ω\subset X$ containing $K$ which admits a complete holomorphic embedding $Ω\to\mathbb{B}^n$ whose image is dense in $\mathbb{B}^n$.

math.CV↗
arXiv2017

Darboux charts around holomorphic Legendrian curves and applications

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,ξ)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds, and we find a holomorphic Darboux chart around any contractible isotropic Stein submanifolds in an arbitrary complex contact manifold.

math.CV↗
arXiv2017

Holomorphic Legendrian curves

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in\mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface $M$ admits a proper holomorphic Legendrian embedding $M\hookrightarrow\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface $M=\mathring M\cup bM$ there exists a topological embedding $M\hookrightarrow \mathbb{C}^{2n+1}$ whose restriction to the interior is a complete holomorphic Legendrian embedding $\mathring M\hookrightarrow \mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold $W$ carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on $W$.

math.CV↗
arXiv2017

New complex analytic methods in the study of non-orientable minimal surfaces in $\mathbb{R}^n$

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to $\mathbb{R}^n$ is a real analytic Banach manifold, obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces, and show general position theorems for non-orientable conformal minimal surfaces in $\mathbb{R}^n$. We also give the first known example of a properly embedded non-orientable minimal surface in $\mathbb{R}^4$; a Mobius strip. All our new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables us to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, we construct proper non-orientable conformal minimal surfaces in $\mathbb{R}^n$ with any given conformal structure, complete non-orientable minimal surfaces in $\mathbb{R}^n$ with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits $n$ hyperplanes of $\mathbb{CP}^{n-1}$ in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in $p$-convex domains of $\mathbb{R}^n$.

math.DG↗
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↗
arXiv2017

Oka properties of ball complements

Let $n>1$ be an integer. We prove that holomorphic maps from Stein manifolds $X$ of dimension $<n$ to the complement $\mathbb{C}^n\setminus L$ of a compact convex set $L\subset\mathbb{C}^n$ satisfy the basic Oka property with approximation and interpolation. If $L$ is polynomially convex then the same holds when $2\dim X < n$. We also construct proper holomorphic maps, immersions and embeddings $X\to\mathbb{C}^n$ with additional control of the range, thereby extending classical results of Remmert, Bishop and Narasimhan.

math.CV↗
arXiv2016

The parametric h-principle for minimal surfaces in $\mathbb R^n$ and null curves in $\mathbb C^n$

Let $M$ be an open Riemann surface. It was proved by Alarcón and Forstnerič (arXiv:1408.5315) that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve $M\to\mathbb C^3$. In this paper, we prove the following much stronger result in this direction: for any $n\geq 3$, the inclusion $ι$ of the space of real parts of nonflat null holomorphic immersions $M\to\mathbb C^n$ into the space of nonflat conformal minimal immersions $M\to \mathbb R^n$ satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equivalence. We prove analogous results for several other related maps, and we describe the homotopy type of the space of all holomorphic immersions $M\to\mathbb C^n$. For an open Riemann surface $M$ of finite topological type, we obtain optimal results by showing that $ι$ and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences.

math.DG↗
arXiv2016

Surjective holomorphic maps onto Oka manifolds

Let $X$ be a connected Oka manifold, and let $S$ be a Stein manifold with $\mathrm{dim} S \geq \mathrm{dim} X$. We show that every continuous map $S\to X$ is homotopic to a surjective strongly dominating holomorphic map $S\to X$. We also find strongly dominating algebraic morphisms from the affine $n$-space onto any compact $n$-dimensional algebraically subelliptic manifold. Motivated by these results, we propose a new holomorphic flexibility property of complex manifolds, the basic Oka property with surjectivity, which could potentially provide another characterization of the class of Oka manifolds.

math.CV↗
arXiv2016

Hyperbolic complex contact structures on $\mathbb{C}^{2n+1}$

In this paper we construct complex contact structures on $\mathbb{C}^{2n+1}$ for any $n\ge 1$ with the property that every holomorphic Legendrian map $\mathbb{C}\to \mathbb{C}^{2n+1}$ is constant. In particular, these contact structures are not globally contactomorphic to the standard complex contact structure on $\mathbb{C}^{2n+1}$.

math.CV↗
arXiv2016

A long $\mathbb C^2$ without holomorphic functions

In this paper we construct for every integer $n>1$ a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of $\mathbb C^n$ (i.e., a long $\mathbb C^n$), but it does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new biholomorphic invariants of a complex manifold $X$, the stable core and the strongly stable core, that are based on the long term behavior of hulls of compact sets with respect to an exhaustion of $X$. We show that every compact polynomially convex set $B\subset \mathbb C^n$ which is the closure of its interior is the strongly stable core of a long $\mathbb C^n$; in particular, biholomorphically nonequivalent sets give rise to nonequivalent long $\mathbb C^n$'s. Furthermore, for any open set $U\subset \mathbb C^n$ there exists a long $\mathbb C^n$ whose stable core is dense in $U$. It follows that for any $n>1$ there is a continuum of pairwise nonequivalent long $\mathbb C^n$'s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long standing open problems.

math.CV↗
arXiv2015

Embedded minimal surfaces in $\mathbb{R}^n$

In this paper, we prove that every confomal minimal immersion of an open Riemann surface into $\mathbb{R}^n$ for $n\ge 5$ can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into $\mathbb{R}^5$. One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to $\mathbb{R}^n$ for any $n\ge 3$ which is also proved in the paper.

math.DG↗
arXiv2015

Minimal hulls of compact sets in $\mathbb R^3$

The main result of this paper is a characterization of the minimal surface hull of a compact set $K$ in $\mathbb R^3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense, and also by $2$-dimensional minimal currents which are limits of Green currents supported by conformal minimal discs. Analogous results are obtained for the null hull of a compact subset of $\mathbb C^3$. We also prove a null hull analogue of the Alexander-Stolzenberg-Wermer theorem on polynomial hulls of compact sets of finite linear measure, and a polynomial hull version of classical Bochner's tube theorem.

math.DG↗
arXiv2015

Every conformal minimal surface in $\mathbb{R}^3$ is isotopic to the real part of a holomorphic null curve

In this paper, we show that for every conformal minimal immersion $u:M\to \mathbb{R}^3$ from an open Riemann surface $M$ to $\mathbb{R}^3$ there exists a smooth isotopy $u_t:M\to\mathbb{R}^3$ $(t\in [0,1])$ of conformal minimal immersions, with $u_0=u$, such that $u_1$ is the real part of a holomorphic null curve $M\to \mathbb{C}^3$ (i.e. $u_1$ has vanishing flux). Furthermore, if $u$ is nonflat then $u_1$ can be chosen to have any prescribed flux and to be complete.

math.DG↗
arXiv2015

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$ which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of $\mathbb{R}^n$ which extend continuously up to the boundary; for $n\ge 5$ we find embeddings with these properties.

math.DG↗
arXiv2015

Noncritical holomorphic functions on Stein spaces

We prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, any closed discrete subset of such a space is the critical locus of a holomorphic function. We also show that for every complex analytic stratification with nonsingular strata on a reduced Stein space there exists a holomorphic function whose restriction to every stratum is noncritical. These result also provide some information on critical loci of holomorphic functions on desingularizations of Stein spaces. In particular, every 1-convex manifold admits a holomorphic function that is noncritical outside the exceptional variety.

math.CV↗