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Ildefonso Castro-Infantes

Publications and source records attributed to Ildefonso Castro-Infantes.

12 recordsLinked to original sources
arXiv2025

Rotational Weingarten surfaces in Lorentz-Minkowski space

We propose a new approach to the study of rotational surfaces in Lorentz-Minkowski space based on the notion of the geometric linear momentum of the generatrix curves with respect to the axes of revolution. This technique allows us to reduce any Weingarten condition on the surface to a first-order ordinary differential equation for the momentum as a function of the distance to the corresponding axis, providing a unified framework that encompasses the three causal types of rotation axes. As a direct application, we classify important families of rotational Weingarten surfaces in this setting, including some linear and quadratic cases. Furthermore, we introduce the non-degenerate quadric surfaces of revolution in Lorentz-Minkowski space and characterize them in terms of a specific cubic Weingarten relation.

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arXiv2024

$\mathrm{CMC\text{-}1}$ surfaces in hyperbolic and de Sitter spaces with Cantor ends

We prove that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits a proper conformal constant mean curvature one ($\mathrm{CMC\text{-}1}$) immersion into hyperbolic $3$-space $\mathbb{H}^3$. Moreover, we obtain that every bordered Riemann surface admits an almost proper $\mathrm{CMC\text{-}1}$ face into de Sitter $3$-space $\mathbb{S}_1^3$, and we show that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits an almost proper $\mathrm{CMC\text{-}1}$ face into $\mathbb{S}_1^3$. These results follow from different uniform approximation theorems for holomorphic null curves in $\mathbb{C}^2 \times \mathbb{C}^*$ that we also establish in this paper.

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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.

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arXiv2023

Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves

We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.

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arXiv2021

Spherical curves whose curvature depends on distance to a great circle

Motivated by a problem posed by David A. Singer in 1999 and by the elastic spherical curves, we study the spherical curves whose curvature is expressed in terms of the distance to a great circle (or from a point). By introducing the notion of spherical angular momentum, we provide new characterizations of some well known curves, like the mentioned elastic curves, spherical catenaries, loxodromic-type spherical curves, the Viviani's curve, and the spherical Archimedean spirals curves. Furthermore, we show that they may be obtained as critical points of some energy curvature functionals. We also find out several new families of spherical curves whose intrinsic equations are expressed in terms of elementary functions or Jacobi elliptic functions, and we are able to get arc length parametrizations of them.

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arXiv2020

Interpolation by complete minimal surfaces whose Gauss map misses two points

Let $M$ be an open Riemann surface and let $Λ\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $Λ$ and whose generalized Gauss map $M\to\mathbb{CP}^{n-1}$, $n\ge 3$, avoids $n$ hyperplanes of $\mathbb{CP}^{n-1}$ located in general position. In case $n=3$, we obtain complete nonflat conformal minimal immersions whose Gauss map $M\to\mathbb{S}^2$ omits two (antipodal) values of the sphere. This result is deduced as a consequence of an interpolation theorem for conformal minimal immersions $M\to\mathbb{R}^n$ into the Euclidean space $\mathbb{R}^n$, $n\ge 3$, with $n-2$ prescribed components.

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arXiv2019

Carleman approximation by conformal minimal immersions and directed holomorphic curves

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theorem. In fact, we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property. Furthermore, the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map. As a consequence we give an approximate solution to a Plateau problem for divergent Jordan curves in the Euclidean spaces.

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arXiv2018

Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

We prove that, given a compact Riemann surface $Σ$ and disjoint finite sets $\varnothing\neq E\subsetΣ$ and $Λ\subsetΣ$, every map $Λ\to \mathbb{R}^3$ extends to a complete conformal minimal immersion $Σ\setminus E\to \mathbb{R}^3$ with finite total curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in $\mathbb{R}^3$ with finite total curvature. To this respect we provide, for each integer $r\ge 1$, a set $A\subset\mathbb{R}^3$ consisting of $12r+3$ points in an affine plane such that if $A$ is contained in a complete nonflat orientable immersed minimal surface $X\colon M\to\mathbb{R}^3$, then the absolute value of the total curvature of $X$ is greater than $4πr$.

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arXiv2018

Curves in Lorentz-Minkowski plane with curvature depending on their position

Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in $\mathbb{L}^2$ whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic) respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper's surface of second kind and for Lorentzian versions of some well known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.

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arXiv2018

Interpolation by conformal minimal surfaces and directed holomorphic curves

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the values of the generalized Gauss map. Furthermore, the interpolating immersions can be chosen to be complete, proper into $\mathbb{R}^n$ if the prescription of values is proper, and one-to-one if $n\ge 5$ and the prescription of values is one-to-one. We may also prescribe the flux map of the examples. We also show analogous results for a large family of directed holomorphic immersions $M\to\mathbb{C}^n$, including null curves.

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arXiv2016

Complete minimal surfaces densely lying in arbitrary domains of $\mathbb{R}^n$

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology. Moreover, we show that every domain in $\mathbb{R}^n$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface. Our method of proof can be adapted to give analogous results for non-orientable minimal surfaces in $\mathbb{R}^n$ $(n\ge 3)$, complex curves in $\mathbb{C}^n$ $(n\ge 2)$, holomorphic null curves in $\mathbb{C}^n$ $(n\ge 3)$, and holomorphic Legendrian curves in $\mathbb{C}^{2n+1}$ $(n\in\mathbb{N})$.

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