arXiv subjects

Alvaro Pampano

Publications and source records attributed to Alvaro Pampano.

At least 19 recordsLinked to original sources

arXiv2026

Renormalized Area of Catenoids in the Hyperbolic Space

We show that the generating curves of non-totally geodesic spherical rotational minimal hypersurfaces (catenoids, for simplicity) of the hyperbolic spaces $\mathbb{H}^{2n+1}$ are $p$-elastic curves for $p=(2n-1)/(2n)$. We employ this variational characterization, together with the Chern--Gauss--Bonnet formulas for locally conformally flat manifolds, to present an explicit expression for the renormalized area of catenoids in terms of hyperelliptic integrals. Further analyzing these special integrals, we show that the renormalized area of catenoids varies continuously from negative infinity to twice the renormalized area of the totally geodesic hypersurfaces $\mathbb{H}^{2n}\subset\mathbb{H}^{2n+1}$. Therefore, we conclude that the renormalized area is not bounded below and that, when $n$ is even, the renormalized area of minimal hypersurfaces in $\mathbb{H}^{2n+1}$ does not have a sign.

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arXiv2026

Renormalized Area of Hypersurfaces in Hyperbolic Spaces

We employ Chen's conformal invariant quantity [8, Theorem 1] in combination with the Chern-Gauss-Bonnet formulas to obtain expressions for the renormalized area of asymptotically minimal hypersurfaces in the $(2n+1)$-dimensional hyperbolic space $\mathbb{H}^{2n+1}$, $n=1,2$. Our results extend Alexakis and Mazzeo's formula for the renormalized area for surfaces in $\mathbb{H}^3$ [1, Proposition 3.1] as well as their relation between the renormalized area of minimal surfaces of $\mathbb{H}^3$ and the Willmore energy of their doubles in $\mathbb{R}^3$ [1, Proposition 8.1] to the non-minimal case and to the higher dimensional case $n=2$. Moreover, we also generalize our results by considering hypersurfaces in $(2n+1)$-dimensional Poincar\'e-Einstein spaces and even-dimensional submanifolds of arbitrary codimension.

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arXiv2025

Hyperbolic Geometry and the Helfrich Functional

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 ${\bf H}^3$. When the membrane surface meets the ideal boundary of hyperbolic space, a modification of the regularized area functional is related to the construction of closed equilibria for the Helfrich functional in ${\bf R}^3$.

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arXiv2024

Closed $p$-Elastic Curves in Spheres of $\mathbb{L}^3$

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 in the hyperbolic plane $\mathbb{H}^2$ these curves exist provided that $p>1$, while in the de Sitter $2$-space $\mathbb{H}_1^2$ the restriction $p<0$ must be satisfied.

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arXiv2024

Stability of Membranes

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 the Helfrich energy for this class of surfaces. The reduced membrane equation also arises as the Euler-Lagrange equation for the area of surfaces under the action of gravity in the three dimensional hyperbolic space. We study the second variation of this functional for a particular example.

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arXiv2023

Integrable Flows on Null Curves in the Anti-de Sitter 3-Space

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 flows in centro-affine geometry. We show that closed stationary solutions of the lower order flow can be explicitly found in terms of periodic solutions of a Lamé equation. In addition, we study the evolution of non-stationary curves arising from a 3-parameter family of periodic solutions of the KdV equation.

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arXiv2023

Closed $1/2$-Elasticae in the Hyperbolic Plane

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 character of their momentum. We prove that closed trajectories arise only when the momentum is a time-like vector. Indeed, for suitable values of the Lagrange multiplier encoding the conservation of the length during the variation, we show the existence of countably many closed trajectories with time-like momentum, which depend on a pair of relatively prime natural numbers.

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arXiv2022

Generalized Elastic Translating Solitons

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 flow by powers of the curvature are shown to be generalized elastic curves. In particular, focusing on the curve shortening flow, we deduce a new variational characterization of the grim reaper curve.

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arXiv2022

Closed 1/2-Elasticae in the 2-Sphere

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 infinitely many closed trajectories which depend on a pair of relatively prime natural numbers. A geometric description of these numbers and the relation with the shape of the corresponding critical trajectories is also given.

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arXiv2022

Instability of Closed $p$-Elastic Curves in $\mathbb{S}^2$

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 relatively prime natural numbers $n$ and $m$ satisfying $m<2n<\sqrt{2}\,m$, there exists a closed spherical $p$-elastic curve with non-constant curvature which winds around a pole $n$ times and closes up in $m$ periods of its curvature. Further, we show that all closed spherical $p$-elastic curves for $p\in(0,1)$ are unstable as critical points of the $p$-elastic energy.

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arXiv2022

Symmetry Breaking Bifurcation of Membranes with Boundary

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 place of working directly with the fourth order membrane equation, it is replaced by a second order reduction found in [16].

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arXiv2022

Closed Biconservative Hypersurfaces in Spheres

We characterise the profile curves of non-CMC biconservative rotational hypersurfaces of space forms $N^n(\rho)$ as $p$-elastic curves, for a suitable rational number $p\in[1/4,1)$ which depends on the dimension $n$ of the ambient space. Analysing the closure conditions of these $p$-elastic curves, we prove the existence of a discrete biparametric family of non-CMC closed (i.e., compact without boundary) biconservative hypersurfaces in $\mathbb{S}^n(\rho)$. None of these hypersurfaces can be embedded in $\mathbb{S}^n(\rho)$.

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arXiv2021

Critical Tori for Mean Curvature Energies in Killing Submersions

We study surface energies depending on the mean curvature in total spaces of Killing submersions, which extend the classical notion of Willmore energy. Based on a symmetry reduction procedure, we construct vertical tori critical for these mean curvature energies. These vertical tori are based on closed curves critical for curvature energy functionals in Riemannian 2-space forms. The binormal evolution of these critical curves in Riemannian 3-space forms generates rotational tori solutions for an ample family of Weingarten surfaces. Therefore, we also introduce some correspondence results between these two types of tori and illustrate their relation.

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arXiv2021

Triharmonic Curves in 3-Dimensional Homogeneous Spaces

We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi-Cartan-Vranceanu spaces.

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arXiv2021

The Euler-Helfrich Functional

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 we studied previously in [23]. It is shown that if a critical surface for this type of functional is axially symmetric, then it satisfies a simpler second order variational problem. Many examples of solutions of this are given.

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arXiv2020

Minimizing Configurations for Elastic Surface Energies with Elastic Boundaries

We study critical surfaces for a surface energy which contains the squared $L^2$ norm of the difference of the mean curvature $H$ and the spontaneous curvature $c_o$, coupled to the elastic energy of the boundary curve. We investigate the existence of equilibria with $H\equiv -c_o$. When $c_o \ge 0$ we characterize those cases where the infimum of this energy is finite for topological annuli and we find the minimizer in the cases that it exists. Results for topological discs are also given.

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arXiv2020

On the Existence of Closed Biconservative Surfaces in Space Forms

Biconservative surfaces of Riemannian 3-space forms $N^3(\rho)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3\kappa_1+\kappa_2=0$ between their principal curvatures $\kappa_1$ and $\kappa_2$. We characterise the profile curves of the non-CMC biconservative surfaces as the critical curves for a suitable curvature energy. Moreover, using this characterisation, we prove the existence of a discrete biparametric family of closed, i.e. compact without boundary, non-CMC biconservative surfaces in the round 3-sphere, $S^3(\rho)$. However, none of these closed surfaces is embedded in $S^3(\rho)$.

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arXiv2019

Totally Biharmonic Hypersurfaces in Space Forms and 3-Dimensional BCV Spaces

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 to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi-Cartan-Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only non-trivial example of a totally biharmonic surface appears in the product space $\mathbb{S}^2(ρ)\times\mathbb{R}$.

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