Antonio Alarcón

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20242026

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math.DG2026

Families of proper minimal surfaces

Antonio Alarcon, Franc Forstneric

Assume that $X$ is a connected, open, oriented smooth surface, $B$ is a compact Euclidean neighbourhood retract, and $J = {J_{b}}_{b \in B}$ is a continuous family of comple…

math.DG2025

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

Antonio Alarcon, Francisco J. Lopez

The Gauss map of a conformal minimal immersion of an open Riemann surface $M$ into $R^{n}$ , $n \geq 3$ , is a holomorphic map $M \to Q^{n - 2} \subset CP^{n - 1}$ . Deno…

math.DG2025

Generic properties of minimal surfaces

Antonio Alarcon, Francisco J. Lopez

Let $M$ be an open Riemann surface and $n \geq 3$ be an integer. In this paper we establish some generic properties (in Baire category sense) in the space of all conformal minimal im…

math.DG2025

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

Antonio Alarcon, Tjasa Vrhovnik

We establish a Mittag-Leffler-type theorem with approximation and interpolation for meromorphic curves $M \to C^{n}$ ( $n \geq 3$ ) directed by Oka cones in $C^{n}$ on an…

math.DG2024

Complete minimal surfaces with Cantor ends in minimally convex domains

Antonio Alarcon

We survey the recent history of the conformal Calabi-Yau problem consisting in determining the complex structures admitted by complete bounded minimal surfaces in $R^{3}$ . M…

math.DG2024

Isotopies of complete minimal surfaces of finite total curvature

Antonio Alarcon, Franc Forstneric, Finnur Larusson

Let $M$ be a Riemann surface biholomorphic to an affine algebraic curve. We show that the inclusion of the space $ℜ NC_{*} (M, C^{n})$ of real parts of nonflat proper…