arXiv subjects

Antonio Bueno

Publications and source records attributed to Antonio Bueno.

At least 19 recordsLinked to original sources

arXiv2026

Invariant $λ$-translators in $\mathbb{S}^2\times\mathbb{R}$

A $λ$-translator in $\mathbb{S}^2\times\mathbb{R}$ is an oriented surface whose mean curvature $H$ satisfies $H=\langle N,\partial_z\rangle+λ$, where $N$ is the unit normal, $\partial_z$ is the vertical Killing vector field tangent to the fibers of the submersion and $λ\in\mathbb{R}$. When $λ=0$ we fall into the class of translators. In this paper, we classify all $λ$-translators that are invariant by a one-parameter group of rotations and by vertical translations of $\mathbb{S}^2\times\mathbb{R}$.

math.DG↗

arXiv2026

Invariant $λ$-translators in the Heisenberg group

We study oriented surfaces in the Heisenberg space $\mathrm{Nil}_3$ whose mean curvature $H$ at each point is $H=\langle N,\partial_z\rangle+λ$, where $N$ is the unit normal, $\partial_z$ is the vertical Killing vector field and $λ\in\mathbb{R}$. These surfaces are known as $λ$-translators and generalize, among others, minimal and positive constant mean curvature surfaces, and also translating solitons of the mean curvature flow. The objective in this paper is to classify $λ$-translators invariant by the following one-parameter groups of isometries of $\mathrm{Nil}_3$: left-translations, rotations and helicoidal motions.

math.DG↗

arXiv2025

Extension of a problem of Euler in $\mathbb{H}^2$ and in $\mathbb{S}^2$

In this paper, we extend the notion of stationary curves with respect to the moment of inertia from a point $N$ in the Euclidean plane $\mathbb{R}^2$ to the case that the ambient space is either the hyperbolic plane $\mathbb{H}^2$ or the sphere $\mathbb{S}^2$. We characterize the critical points of this energy in terms of the curvature of the curve and the distance to $N$. In $\mathbb{H}^2$, we prove that the only closed stationary curves are circles centered at $N$. In $\mathbb{S}^2$, we estimate the value of $α$ for closed curves according to the hemisphere of $\mathbb{S}^2$ in which the curve lies. In addition, we find the first integrals of the ODEs that describe the parametrizations of stationary curves in both ambient spaces. Finally, we consider the energy minimization problem for curves connecting two points collinear with $N$, in particular solving the case of geodesics.

math.DG↗

arXiv2024

On the stability of Killing cylinders in hyperbolic space

In this paper we study the stability of a Killing cylinder in hyperbolic 3-space when regarded as a capillary surface for the partitioning problem. In contrast with the Euclidean case, we consider a variety of totally umbilical support surfaces, including horospheres, totally geodesic planes, equidistant surfaces and round spheres. In all of them, we explicitly compute the Morse index of the corresponding eigenvalue problem for the Jacobi operator. We also address the stability of compact pieces of Killing cylinders with Dirichlet boundary conditions when the boundary is formed by two fixed circles, exhibiting an analogous to the Plateau-Rayleigh instability criterion for Killing cylinders in the Euclidean space. Finally, we prove that the Delaunay surfaces can be obtained by bifurcating Killing cylinders supported on geodesic planes.

math.DG↗

arXiv2024

Stability of cylinders in $\mathbb{E}(κ,τ)$ homogeneous spaces

We extend the classical Plateau-Rayleigh instability criterion in the $\mathbb{E}(κ,τ)$ spaces. We prove the existence of a positive number $L_0>0$ such that if a truncated circular cylinder of radius $ρ$ in $\mathbb{E}(κ,τ)$ has length $L>L_0$ then it is unstable. This number $L_0$ depends on $κ$, $τ$ and $ρ$. The value $L_0$ is sharp under axially-symmetric variations of the surface. We also extend this result for the partitioning problem in $\mathbb{E}(κ,τ)$.

math.DG↗

arXiv2024

The class of grim reapers in $\mathbb{H}^2\times\mathbb{R}$

We study translators of the mean curvature flow in the product space $\h^2\times\r$. In $\h^2\times\r$ there are three types of translations: vertical translations due to the factor $\r$ and parabolic and hyperbolic translations from $\h^2$. A grim reaper in $\h^2\times\r$ is a translator invariant by a one-parameter group of translations. The variety of translators and translations in $\h^2\times\r$ makes that the family of grim reapers particularly rich. In this paper we give a full classification of the grim reapers of $\h^2\times\r$ with a description of their geometric properties. In some cases, we obtain explicit parametrizations of the surfaces.

math.DG↗

arXiv2024

Invariant $λ$-translators in Lorentz-Minkowski space

Given $λ\in\mathbb{R}$ and $\textbf{v}\in\mathbb{L}^3$, a $λ$-translator with velocity $\textbf{v}$ is an immersed surface in $\mathbb{L}^3$ whose mean curvature satisfies $H=\langle N,\textbf{v}\rangle+λ$, where $N$ is a unit normal vector field. When $λ=0$, we fall into the class of translating solitons of the mean curvature flow. In this paper we study $λ$-translators in $\mathbb{L}^3$ that are invariant under a 1-parameter group of translations and rotations. The former are cylindrical surfaces and explicit parametrizations are found, distinguishing on the causality of both the ruling direction and the $λ$-translators. In the case of rotational $λ$-translators we distinguish between spacelike and timelike rotations and exhibit the qualitative properties of rotational $λ$-translators by analyzing the non-linear autonomous system fulfilled by the coordinate functions of the generating curves.

math.DG↗

arXiv2024

Horo-shrinkers in the hyperbolic space

A surface $Σ$ in the hyperbolic space $\h^3$ is called a horo-shrinker if its mean curvature $H$ satisfies $H=\langle N,\partial_z\rangle$, where $(x,y,z)$ are the coordinates of $\h^3$ in the upper half-space model and $N$ is the unit normal of $Σ$. In this paper we study horo-shrinkers invariant by one-parameter groups of isometries of $\h^3$ depending if these isometries are hyperbolic, parabolic or spherical. We characterize totally geodesic planes as the only horo-shrinkers invariant by a one-parameter group of hyperbolic translations. The grim reapers are defined as the horo-shrinkers invariant by a one-parameter group of parabolic translations. We describe the geometry of the grim reapers proving that they are periodic surfaces. In the last part of the paper, we give a complete classification of horo-shrinkers invariant by spherical rotations, distinguishing if the surfaces intersect or not the rotation axis.

math.DG↗

arXiv2024

A new family of translating solitons in hyperbolic space

If $ξ$ is a Killing vector field of the hyperbolic space $\h^3$ whose flow are parabolic isometries, a surface $Σ\subset\h^3$ is a $ξ$-translator if its mean curvature $H$ satisfies $H=\langle N,ξ\rangle$, where $N$ is the unit normal of $Σ$. We classify all $ξ$-translators invariant by a one-parameter group of rotations of $\h^3$, exhibiting the existence of a new family of grim reapers. We use these grim reapers to prove the non-existence of closed $ξ$-translators.

math.DG↗

arXiv2023

Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map

Given a $C^1$ function $\mathcal{H}$ defined in the unit sphere $\mathbb{S}^2$, an $\mathcal{H}$-surface $M$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_M$ satisfies $H_M(p)=\mathcal{H}(N_p)$, $p\in M$, where $N$ is the Gauss map of $M$. Given a closed simple curve $Γ\subset\mathbb{R}^3$ and a function $\mathcal{H}$, in this paper we investigate the geometry of compact $\mathcal{H}$-surfaces spanning $Γ$ in terms of $Γ$. Under mild assumptions on $\mathcal{H}$, we prove non-existence of closed $\mathcal{H}$-surfaces, in contrast with the classical case of constant mean curvature. We give conditions on $\mathcal{H}$ that ensure that if $Γ$ is a circle, then $M$ is a rotational surface. We also establish the existence of estimates of the area of $\mathcal{H}$-surfaces in terms of the height of the surface.

math.DG↗

arXiv2023

The Plateau-Rayleigh instability of translating $λ$-solitons

Given a unit vector $\textbf{v}\in\mathbb{R}^3$ and $λ\in\mathbb{R}$, a translating $λ$-soliton is a surface in $\mathbb{R}^3$ whose mean curvature $H$ satisfies $H=\langle N,\textbf{v}\rangle+λ,\ |\textbf{v}|=1$, where $N$ is the Gauss map of the surface. In this paper, we extend the phenomenon of instability of Plateau-Rayleigh for translating $λ$-solitons of cylindrical type, proving that long pieces of these surfaces are unstable. We will provide explicit bounds on the length of these surfaces. It will be also proved that if a translating $λ$-soliton is a graph, then it is a minimizer of the weighted area in a suitable class of surfaces with the same boundary and the same weighted volume.

math.DG↗

arXiv2022

Surfaces of prescribed linear Weingarten curvature in $\mathbb{R}^3$

Given $a,b\in\mathbb{R}$ and $Φ\in C^1(\mathbb{S}^2)$, we study immersed oriented surfaces $Σ$ in the Euclidean 3-space $\mathbb{R}^3$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=Φ(N)$, where $N:Σ\rightarrow\mathbb{S}^2$ is the Gauss map. This theory widely generalize some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $Φ$, we exhibit a classification result for rotational surfaces in the case that the underlying fully nonlinear PDE that governs these surfaces is elliptic or hyperbolic.

math.DG↗

arXiv2022

Rotational surfaces of prescribed Gauss curvature in $\mathbb{R}^3$

We study rotational surfaces in Euclidean 3-space whose Gauss curvature is given as a prescribed function of its Gauss map. By means of a phase plane analysis and under mild assumptions on the prescribed function, we generalize the classification of rotational surfaces of constant Gauss curvature; exhibit examples that cannot exist in the constant Gauss curvature case; and analyze the asymptotic behavior of strictly convex graphs. We also prove the existence of singular radial solutions intersecting orthogonally the axis of rotation.

math.DG↗

arXiv2022

Radial solutions for equations of Weingarten type

In this paper we study the linear Weingarten equation defined by the fully non-linear PDE $$a\, \mbox{div}\frac{Du}{\sqrt{1+|Du|^2}}+b\, \frac{\mbox{det}D^2u}{(1+|Du|^2)^2}=ϕ\left(\frac{1}{\sqrt{1+|Du|^2}}\right)$$ in a domain $Ω\subset\mathbb{R}^2$, where $ϕ\in C^1([-1,1])$ and $a,b\in\mathbb{R}$. We approach the existence of radial solutions when $Ω$ is a disk of small radius, giving an affirmative answer when the PDE is of elliptic type. In the hyperbolic case we show that no radial solution exists, while in the parabolic case we find explicitly all the solutions. Finally, in the elliptic case we prove uniqueness and symmetry results concerning the Dirichlet problem of such equation.

math.AP↗

arXiv2021

Delaunay surfaces of prescribed mean curvature in $\mathrm{Nil}_3$ and $\widetilde{SL_2}(\mathbb{R})$

We obtain a classification result for rotational surfaces in the Heisenberg space and the universal cover of the special linear group, whose mean curvature is given as a prescribed $C^1$ function depending on their angle function. We show that these surfaces behave like the Delaunay surfaces of constant mean curvature, under some assumptions on the prescribed function. In contrast with the constant mean curvature case, we exhibit the existence of rotational, embedded tori, providing counterexamples of the Alexandrov problem for this class of immersed surfaces.

math.DG↗

arXiv2020

Surfaces with prescribed mean curvature in $\mathbb{H}^2\times\mathbb{R}$

In this paper we study rotational surfaces in the space $\mathbb{H}^2\times\mathbb{R}$ whose mean curvature is given as a prescribed function of their angle function. These surfaces generalize, among others, the ones of constant mean curvature and the translating solitons of the mean curvature flow. Using a phase plane analysis we construct entire rotational graphs, catenoid-type surfaces, and exhibit a classification result when the prescribed function is linear.

math.DG↗

arXiv2019

Invariant hypersurfaces with linear prescribed mean curvature

Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely related with the theory of manifolds with density, since their weighted mean curvature in the sense of Gromov is constant. In this paper we obtain explicit parametrizations of constant curvature hypersurfaces, and also give a classification of rotationally invariant hypersurfaces.

math.DG↗

arXiv2019

A Delaunay-type classification result for prescribed mean curvature surfaces in $\mathbb{M}^2(κ)\times\mathbb{R}$

The purpose of this paper is to study immersed surfaces in the product spaces $\mathbb{M}^2(κ)\times\mathbb{R}$, whose mean curvature is given as a $C^1$ function depending on their angle function. This class of surfaces extends widely, among others, the well-known theory of surfaces with constant mean curvature. In this paper we give necessary and sufficient conditions for the existence of prescribed mean curvature spheres, and we describe complete surfaces of revolution proving that they behave as the Delaunay surfaces of CMC type.

math.DG↗