arXiv subjects

Stefano Montaldo

Publications and source records attributed to Stefano Montaldo.

At least 19 recordsLinked to original sources

arXiv2026

Existence and stability of weak critical points of $r$-energy functionals

The main aim of this paper is to prove the existence of certain proper weakly $r$-harmonic ($ES-r$-harmonic) maps. We construct critical points which belong to a family of rotationally symmetric maps $φ_a : B^n \to \mathbb{S}^n$, where $B^n$ and $\mathbb{S}^n$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. We find that the existence of solutions within this family is restricted to specific dimensions $n$. Next, we prove that our critical points are \textit{unstable}. In the course of this analysis we point out some specific differences between the $r$-harmonic and the $ES-r$-harmonic cases when $r \geq 4$. Next, we analyse two variants of the problem. First, we replace the target manifold $\mathbb{S}^n$ with a rotationally symmetric ellipsoid $E^n(b)$ and establish the existence of proper weakly biharmonic maps for all $n \geq 5$, as well as proper weakly triharmonic maps for all $n \geq 7$. Finally, we study a similar problem replacing the domain $B^n$ with a suitable warped product manifold.

math.DG↗

arXiv2025

Biconservative Weingarten surfaces with flat normal bundle in $N^4 (ε)$

In this paper, we extend our investigation of the class of biconservative surfaces with non-constant mean curvature in 4-dimensional space forms $N^4(ε)$. Specifically, we focus on biconservative surfaces with non-parallel normalized mean curvature vector fields (non-PNMC) that have flat normal bundles and are Weingarten. In our initial result we obtain the compatibility conditions for this class of biconservative surfaces in terms of an ODE system. Subsequently, by prescribing the flat connection in the normal bundle, we prove an existence result for the considered class of biconservative surfaces. Furthermore, we determine all non-PNMC biconservative Weingarten surfaces with flat normal bundles that either exhibit a particular form of the shape operator in the direction of the mean curvature vector field or have constant Gaussian curvature $K = ε$. Finally, we prove that such surfaces cannot be biharmonic.

math.DG↗

arXiv2025

Polyharmonic curves in semi-Riemannian manifolds

Let $(M^m_t,g)$ be a semi-Riemannian manifold of dimension $m$ with a non-degenerate metric of \textit{index} $t$, $m\geq 2$, $1 \leq t \leq m-1$. The main aim of this paper is to investigate the existence of Frenet curves in $(M^m_t,g)$ which are polyharmonic of order $r$, shortly, $r$-harmonic. We shall focus primarily on the cases that the ambient space is a semi-Riemannian space form $N^m_t(c)$ of sectional curvature $c$, a ruled Lorentzian surface or a suitable, possibly warped, product space. We shall obtain existence, non-existence and classification results.

math.DG↗

arXiv2025

An Invitation to Biharmonic and Biconservative Submanifolds

This note is based on a lecture delivered by the author at the Second Conference on Differential Geometry, held in Fez in October 2024. It offers an accessible introduction to biharmonic and biconservative submanifolds, exploring the motivations for their study and highlighting some key facts and open problems in the field.

math.DG↗

arXiv2024

Polyharmonic helices

The main aim of this paper is to investigate the existence of Frenet helices which are polyharmonic of order $r$, shortly, $r$-harmonic. We shall obtain existence, non-existence and classification results. More specifically, we obtain a complete classification of proper $r$-harmonic helices into the $3$-dimensional solvable Lie group Sol$_3$. Next, we investigate the existence of proper $r$-harmonic helices into Bianchi-Cartan-Vranceanu spaces and, in this context, we find new examples. Finally, we shall establish some non-existence results both for Frenet curves and Frenet helices of order $n \geq 4$ when the ambient space is the Euclidean sphere $\s^m$.

math.DG↗

arXiv2024

Triharmonic curves in the 3-dimensional Sol space

The main aim of this paper is to study triharmonic curves in the 3-dimensional homogeneous space Sol. In the first part of the paper we shall obtain a complete classification of proper triharmonic curves with constant geodesic curvature and torsion. In the final section we shall show that these triharmonic curves form a constant angle with a suitable Killing field of constant length along the curve.

math.DG↗

arXiv2023

Polyharmonic surfaces in $3$-dimensional homogeneous spaces

In the first part of this paper we shall classify proper triharmonic isoparametric surfaces in 3-dimensional homogeneous spaces (Bianchi-Cartan-Vranceanu spaces, shortly BCV-spaces). We shall also prove that triharmonic Hopf cylinders are necessarily CMC. In the last section we shall determine a complete classification of CMC r-harmonic Hopf cylinders in BCV-spaces, r>=3. This result ensures the existence, for suitable values of r, of an ample family of new examples of r-harmonic surfaces in BCV-spaces.

math.DG↗

arXiv2022

Closed Biconservative Hypersurfaces in Spheres

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. 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 $\mathbb{S}^n(ρ)$. None of these hypersurfaces can be embedded in $\mathbb{S}^n(ρ)$.

math.DG↗

arXiv2022

On the second variation of the biharmonic Clifford torus in S^4

The flat torus ${\mathbb T}={\mathbb S}^1\left (\frac{1}{2} \right ) \times {\mathbb S}^1\left (\frac{1}{2} \right )$ admits a proper biharmonic isometric immersion into the unit $4$-dimensional sphere ${\mathbb S}^4$ given by $Φ=i \circ φ$, where $φ:{\mathbb T} \to {\mathbb S}^3(\frac{1}{\sqrt 2})$ is the minimal Clifford torus and $i:{\mathbb S}^3(\frac{1}{\sqrt 2}) \to {\mathbb S}^4$ is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion $Φ$. After, we shall study in the detail the kernel of the generalised Jacobi operator $I_2^Φ$. We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper we shall analyse the specific contribution of $φ$ to the biharmonic index and nullity of $Φ$. In this context, we shall study a more general composition $\tildeΦ=\tildeφ \circ i$, where $\tildeφ: M^m \to {\mathbb S}^{n-1}(\frac{1}{\sqrt 2})$, $ m \geq 1$, $n \geq {3}$, is a minimal immersion and $i:{\mathbb S}^{n-1}(\frac{1}{\sqrt 2}) \to {\mathbb S}^n$ is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation of $\tildeΦ$ is nonnegatively defined on $\mathcal{C}\big (\tildeφ^{-1}T{\mathbb S}^{n-1}\big )$. Then we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the $p$-harmonic index and nullity of $φ$. In the final section we compare our general results with those which can be deduced from the study of the equivariant second variation.

math.DG↗

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.

math.DG↗

arXiv2021

Interpreting the outcomes of research assessments: a geometrical approach

Research evaluations and comparison of the assessments of academic institutions (scientific areas, departments, universities etc.) are among the major issues in recent years in higher education systems. One method, followed by some national evaluation agencies, is to assess the research quality by the evaluation of a limited number of publications in a way that each publication is rated among $n$ classes. This method produces, for each institution, a distribution of the publications in the $n$ classes. In this paper we introduce a natural geometric way to compare these assessments by introducing an ad hoc distance from the performance of an institution to the best possible achievable assessment. Moreover, to avoid the methodological error of comparing non-homogeneous institutions, we introduce a {\em geometric score} based on such a distance. The latter represents the probability that an ideal institution, with the same configuration as the one under evaluation, performs worst. We apply our method, based on the geometric score, to rank, in two specific scientific areas, the Italian universities using the results of the evaluation exercise VQR 2011-2014.

cs.DL↗

arXiv2021

Unique continuation properties for polyharmonic maps between Riemannian manifolds

Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic case: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on a open subset, then they agree everywhere; (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.

math.DG↗

arXiv2020

On the Existence of Closed Biconservative Surfaces in Space Forms

Biconservative surfaces of Riemannian 3-space forms $N^3(ρ)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3κ_1+κ_2=0$ between their principal curvatures $κ_1$ and $κ_2$. 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, $S^3(ρ)$. However, none of these closed surfaces is embedded in $S^3(ρ)$.

math.DG↗

arXiv2020

On the stability of the equator map for higher order energy functionals

Let $B^n\subset {\mathbb R}^{n}$ and ${\mathbb S}^n\subset {\mathbb R}^{n+1}$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. The \textit{extrinsic $k$-energy functional} is defined on the Sobolev space $W^{k,2}\left (B^n,{\mathbb S}^n \right )$ as follows: $E_{k}^{\rm ext}(u)=\int_{B^n}|Δ^s u|^2\,dx$ when $k=2s$, and $E_{k}^{\rm ext}(u)=\int_{B^n}|\nabla Δ^s u|^2\,dx$ when $k=2s+1$. These energy functionals are a natural higher order version of the classical extrinsic bienergy, also called Hessian energy. The equator map $u^*: B^n \to {\mathbb S}^n$, defined by $u^*(x)=(x/|x|,0)$, is a critical point of $E_{k}^{\rm ext}(u)$ provided that $n \geq 2k+1$. The main aim of this paper is to establish necessary and sufficient conditions on $k$ and $n$ under which $u^*: B^n \to {\mathbb S}^n$ is minimizing or unstable for the extrinsic $k$-energy.

math.DG↗

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 $\mathbb{S}^2(ρ)\times\mathbb{R}$.

math.DG↗

arXiv2019

Higher order energy functionals

The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called $ES-r$-energy functionals $ E_r^{ES}(φ)=(1/2)\int_{M}\,|(d^*+d)^r (φ)|^2\,dV$, where $ φ:M \to N$ is a map between two Riemannian manifolds. In the initial part of this paper we shall clarify some relevant issues about the definition of an $ES-r$-harmonic map, i.e, a critical point of $ E_r^{ES}(φ)$. That seems important to us because in the literature other higher order energy functionals have been studied by several authors and consequently some recent examples need to be discussed and extended: this shall be done in the first two sections of this work, where we obtain the first examples of proper critical points of $E_r^{ES}(φ)$ when $N={\mathbb S}^m$ $(r \geq4,\, m\geq3)$, and we also prove some general facts which should be useful for future developments of this subject. Next, we shall compute the Euler-Lagrange system of equations for $E_r^{ES}(φ)$ for $r=4$. We shall apply this result to the study of maps into space forms and to rotationally symmetric maps: in particular, we shall focus on the study of various family of conformal maps. In Section 4, we shall also show that, even if $2 r > \dim M$, the functionals $ E_r^{ES}(φ)$ may not satisfy the classical Palais-Smale Condition (C). In the final part of the paper we shall study the second variation and compute index and nullity of some significant examples.

math.DG↗

arXiv2018

Biharmonic constant mean curvature surfaces in Killing submersions

A $3$-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part, we also classify the proper biharmonic Hopf cylinders in a Killing submersion.

math.DG↗

arXiv2017

A note on the Almansi property

The first goal of this note is to study the Almansi property on an m-dimensional model in the sense of Greene and Wu and, more generally, in a Riemannian geometric setting. In particular, we shall prove that the only model on which the Almansi property is verified is the Euclidean space R^m. In the second part of the paper we shall study Almansi's property and biharmonicity for functions which depend on the distance from a given submanifold. Finally, in the last section we provide an extension to the semi-Euclidean case R^{p,q} which includes the proof of the classical Almansi property in R^m as a special instance.

math.DG↗