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Antonio Alarcon

Publications and source records attributed to Antonio Alarcon.

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 Hölder class $\mathscr{C}^α$ for some $0<α<1$. 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↗
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

Generic properties of minimal surfaces

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. In this paper we establish some generic properties (in Baire category sense) in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology, pointing out that a generic such immersion is chaotic in many ways. For instance, we show that a generic conformal minimal immersion $u\colon M\to \mathbb{R}^n$ is non-proper, almost proper, and $g$-complete with respect to any given Riemannian metric $g$ in $\mathbb{R}^n$. Further, its image $u(M)$ is dense in $\mathbb{R}^n$ and disjoint from $\mathbb{Q}^3\times \mathbb{R}^{n-3}$, and has infinite area, infinite total curvature, and unbounded curvature on every open set in $\mathbb{R}^n$. In case $n=3$, we also prove that a generic conformal minimal immersion $M\to\mathbb{R}^3$ has infinite index of stability on every open set in $\mathbb{R}^3$.

math.DG↗
arXiv2025

On the Gauss map assignment for minimal surfaces and the Osserman curvature estimate

The Gauss map of a conformal minimal immersion of an open Riemann surface $M$ into $\mathbb{R}^n$, $n\ge 3$, is a holomorphic map $M\to{\bf Q}^{n-2}\subset \mathbb{CP}^{n-1}$. Denote by ${\rm CMI}_{\rm full}(M,\mathbb{R}^n)$ and $\mathscr{O}_{\rm full}(M,{\bf Q}^{n-2})$ the spaces of full conformal minimal immersions $M\to\mathbb{R}^n$ and full holomorphic maps $M\to{\bf Q}^{n-2}$, respectively, endowed with the compact-open topology. In this paper we show that the Gauss map assignment $\mathscr{G}:{\rm CMI}_{\rm full}(M,\mathbb{R}^n)\to \mathscr{O}_{\rm full}(M,{\bf Q}^{n-2})$, taking a full conformal minimal immersion to its Gauss map, is an open map. This implies, in view of a result of Forstneric and the authors, that $\mathscr{G}$ is a quotient map. The same results hold for the map $(\mathscr{G},Flux):{\rm CMI}_{\rm full}(M,\mathbb{R}^n)\to \mathscr{O}_{\rm full}(M,{\bf Q}^{n-2})\times H^1(M,\mathbb{R}^n)$, where $Flux:{\rm CMI}_{\rm full}(M,\mathbb{R}^n)\to H^1(M,\mathbb{R}^n)$ is the flux assignment. As application, we establish that the set of maps $G\in \mathscr{O}_{\rm full}(M,{\bf Q}^{n-2})$ such that the family $\mathscr{G}^{-1}(G)$ of all minimal surfaces in $\mathbb{R}^n$ with the Gauss map $G$ satisfies the classical Osserman curvature estimate, is meagre in the space of holomorphic maps $M\to {\bf Q}^{n-2}$.

math.DG↗
arXiv2025

The Mittag-Leffler theorem for proper minimal surfaces and directed meromorphic curves

We establish a Mittag-Leffler-type theorem with approximation and interpolation for meromorphic curves $M\to \mathbb{C}^n$ ($n\geq 3$) directed by Oka cones in $\mathbb{C}^n$ on any open Riemann surface $M$. We derive a result of the same type for proper conformal minimal immersions $M\to \mathbb{R}^n$. This includes interpolation in the poles and approximation by embeddings, the latter if $n\ge 5$ in the case of minimal surfaces. As applications, we show that complete minimal ends of finite total curvature in $\mathbb{R}^5$ are generically embedded, and characterize those open Riemann surfaces which are the complex structure of a proper minimal surface in $\mathbb{R}^3$ of weak finite total curvature.

math.DG↗
arXiv2025

Holomorphic null curves in the special linear group

In this paper we develop the theory of approximation for holomorphic null curves in the special linear group ${\rm SL}_2(\mathbb{C})$. In particular, we establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for the family of holomorphic null immersions $M\to{\rm SL}_2(\mathbb{C})$ from any open Riemann surface $M$. Our results include jet interpolation of Weierstrass type and approximation by embeddings, as well as global conditions on the approximating curves. As application, we show that every open Riemann surface admits a proper holomorphic null embedding into ${\rm SL}_2(\mathbb{C})$, and hence also a proper conformal immersion of constant mean curvature $1$ into hyperbolic 3-space. This settles a problem posed by Alarcon and Forstneric in 2015.

math.DG↗
arXiv2025

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↗
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

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

Complete minimal surfaces with Cantor ends in minimally convex domains

We survey the recent history of the conformal Calabi-Yau problem consisting in determining the complex structures admitted by complete bounded minimal surfaces in $\mathbb{R}^3$. Moreover, we prove that for any minimally convex domain $Ω$ in $\mathbb{R}^3$ and any compact Riemann surface $R$ there is a Cantor set $C$ in $R$ whose complement $R\setminus C$ is the complex structure of a complete proper minimal surface in $Ω$.

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↗
arXiv2024

Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure

We prove that every open Riemann surface $M$ is the complex structure of a complete surface of constant mean curvature 1 (CMC-1) in the 3-dimensional hyperbolic space $\mathbb{H}^3$. We go further and establish a jet interpolation theorem for complete conformal CMC-1 immersions $M\to \mathbb{H}^3$. As a consequence, we show the existence of complete densely immersed CMC-1 surfaces in $\mathbb{H}^3$ with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in $\mathbb{C}^2\times\mathbb{C}^*$ which is also established in this paper.

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 $Φ$ of $\mathbb{C}^n$ such that the image foliation $Φ_*\mathcal{F}$ is $\mathfrak{g}$-complete. The analogous result is obtained on every Stein manifold with Varolin's density property.

math.CV↗
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↗
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 $\mathbb{CP}^3$ 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↗