Álvaro Pámpano
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View article: Hyperbolic Geometry and the Helfrich Functional
Hyperbolic Geometry and the Helfrich Functional Open
The Helfrich model is a fundamental tool for determining the morphology of biological membranes. We relate the geometry of an important class of its equilibria to the geometry of sessile and pendant drops in the hyperbolic space ${\bf H}^3…
View article: Geometric Transformations on Null Curves in the Anti-de Sitter 3-Space
Geometric Transformations on Null Curves in the Anti-de Sitter 3-Space Open
We provide a geometric transformation on null curves in the anti-de Sitter 3-space (AdS) which induces the Backlund transformation for the KdV equation. In addition, we show that this geometric transformation satisfies a suitable permutabi…
View article: Axially Symmetric Helfrich Spheres
Axially Symmetric Helfrich Spheres Open
Smooth axially symmetric Helfrich topological spheres are either round or else they must satisfy a second order equation known as the reduced membrane equation [17]. In this paper, we show that, conversely, axially symmetric closed genus z…
View article: Integrable flows on null curves in the Anti-de Sitter 3-space
Integrable flows on null curves in the Anti-de Sitter 3-space Open
We formulate integrable flows related to the Korteweg–De Vries (KdV) hierarchy on null curves in the anti-de Sitter 3-space ( ). Exploiting the specific properties of the geometry of , we analyze their interrelationships with Pinkall…
View article: Closed $p$-Elastic Curves in Spheres of $\mathbb{L}^3$
Closed $p$-Elastic Curves in Spheres of $\mathbb{L}^3$ Open
For every $p\in\mathbb{R}$, we study $p$-elastic curves in the hyperbolic plane $\mathbb{H}^2$ and in the de Sitter $2$-space $\mathbb{H}_1^2$. We analyze the existence of closed $p$-elastic curves with nonconstant curvature showing that i…
View article: Stability of Membranes
Stability of Membranes Open
In [12], the authors studied a particular class of equilibrium solutions of the Helfrich energy which satisfy a second order condition called the reduced membrane equation. In this paper we develop and apply a second variation formula for …
View article: Geometric Transformations on Null Curves in the Anti-de Sitter 3-Space
Geometric Transformations on Null Curves in the Anti-de Sitter 3-Space Open
We provide a geometric transformation on null curves in the anti-de Sitter 3-space (AdS) which induces the Bäcklund transformation for the KdV equation. In addition, we show that this geometric transformation satisfies a suitable permutabi…
View article: Integrable Flows on Null Curves in the Anti-de Sitter 3-Space
Integrable Flows on Null Curves in the Anti-de Sitter 3-Space Open
We formulate integrable flows related to the KdV hierarchy on null curves in the anti-de Sitter 3-space (${\rm AdS}$). Exploiting the specific properties of the geometry of ${\rm AdS}$, we analyze their interrelationships with Pinkall flow…
View article: Closed $1/2$-Elasticae in the Hyperbolic Plane
Closed $1/2$-Elasticae in the Hyperbolic Plane Open
We study critical trajectories in the hyperbolic plane for the $1/2$-Bernoulli's bending energy with length constraint. Critical trajectories with periodic curvature are classified into three different types according to the causal charact…
View article: On the existence of closed biconservative surfaces in space forms
On the existence of closed biconservative surfaces in space forms Open
Biconservative surfaces of Riemannian 3-space forms $N^3(\\rho)$, are either\nconstant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces\nverifying the relation $3\\kappa_1+\\kappa_2=0$ between their principal\ncurvatu…
View article: Generalized Elastic Translating Solitons
Generalized Elastic Translating Solitons Open
We study translating soliton solutions to the flow by powers of the curvature of curves in the plane. We characterize these solitons as critical curves for functionals depending on the curvature. More precisely, translating solitons to the…
View article: Instability of Closed $p$-Elastic Curves in $\mathbb{S}^2$
Instability of Closed $p$-Elastic Curves in $\mathbb{S}^2$ Open
For $p\in\mathbb{R}$, we show that non-circular closed $p$-elastic curves in $\mathbb{S}^2$ exist only when $p=2$, in which case they are classical elastic curves, or when $p\in(0,1)$. In the latter case, we prove that for every pair of re…
View article: Symmetry Breaking Bifurcation of Membranes with Boundary
Symmetry Breaking Bifurcation of Membranes with Boundary Open
We use a bifurcation theory due to Crandall and Rabinowitz to show the existence of a symmetry breaking bifurcation of a specific one parameter family of axially symmetric disc type solutions of a membrane equation with fixed boundary. In …
View article: A relation between cylindrical critical points of Willmore-type energies, weighted areas and vertical potential energies
A relation between cylindrical critical points of Willmore-type energies, weighted areas and vertical potential energies Open
This paper considers the energies of three different physical scenarios and obtains relations between them in a particular case. The first family of energies consists of the Willmore-type energies involving the integral of powers of the me…
View article: Closed 1/2-Elasticae in the 2-Sphere
Closed 1/2-Elasticae in the 2-Sphere Open
We study critical trajectories in the sphere for the $1/2$-Bernoulli's bending functional with length constraint. For every Lagrange multiplier encoding the conservation of the length during the variation, we show the existence of infinite…
View article: Closed Biconservative Hypersurfaces in Spheres
Closed Biconservative Hypersurfaces in Spheres Open
We characterise the profile curves of non-CMC biconservative rotational hypersurfaces of space forms $N^n(ρ)$ as $p$-elastic curves, for a suitable rational number $p\in[1/4,1)$ which depends on the dimension $n$ of the ambient space. Anal…
View article: Stationary soap films with vertical potentials
Stationary soap films with vertical potentials Open
We classify cylindrical surfaces in the Euclidean space whose mean curvature is a $n$th-power of the distance to a reference plane. The generating curves of these surfaces, called $n$-elastic curves, have a variational characterization as …
View article: On p-Willmore Disks with Boundary Energies
On p-Willmore Disks with Boundary Energies Open
We consider an energy functional on surface immersions which includes contributions from both boundary and interior. Inspired by physical examples, the boundary is modeled as the center line of a generalized Kirchhoff elastic rod, while th…
View article: The Euler-Helfrich Functional
The Euler-Helfrich Functional Open
We investigate equilibrium configurations for surface energies which contain the squared $L^2$ norm of the difference of the mean curvature H and the spontaneous curvature $c_o$ coupled with the elastic energy of the boundary curve, which …
View article: Totally biharmonic hypersurfaces in space forms and 3-dimensional BCV spaces
Totally biharmonic hypersurfaces in space forms and 3-dimensional BCV spaces Open
A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us…
View article: Regarding the Euler-Plateau Problem with Elastic Modulus
Regarding the Euler-Plateau Problem with Elastic Modulus Open
We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is …
View article: Classification of rotational surfaces with constant skew curvature in 3-space forms
Classification of rotational surfaces with constant skew curvature in 3-space forms Open
In this paper, we classify the rotational surfaces with constant skew curvature in $3$-space forms. We also give a variational characterization of the profile curves of these surfaces as critical points of a curvature energy involving the …
View article: Rotational surfaces of constant astigmatism in space forms
Rotational surfaces of constant astigmatism in space forms Open
A surface in a Riemannian space is called of constant astigmatism if the difference between the principal radii of curvatures at each point is a constant function. In this paper we give a classification of all rotational surfaces of consta…
View article: Rotational surfaces of constant astigmatism in space forms
Rotational surfaces of constant astigmatism in space forms Open
A surface in a Riemannian space is called of constant astigmatism if the difference between the principal radii of curvatures at each point is a constant function. In this paper we give a classification of all rotational surfaces of consta…
View article: Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures
Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures Open
We classify all rotational surfaces in Euclidean space whose principal curvatures κ 1 and κ 2 satisfy the linear relation , where a and b are two constants. As a consequence of this classification, we find closed (embedded and not embedded…
View article: Classification of Planar Anisotropic Elasticae
Classification of Planar Anisotropic Elasticae Open
We classify the anisotropic elastic curves modulo rescaling and quasi-rotation depending on one parameter for an ample family of anisotropic functionals. Several illustrations of this classification are shown at the end.