arXiv subjects

Cezar Oniciuc

Publications and source records attributed to Cezar Oniciuc.

At least 19 recordsLinked to original sources

arXiv2026

On conformal biharmonic maps and hypersurfaces

In this article we initiate a thorough geometric study of the conformal bienergy functional which consists of the standard bienergy augmented by two additional curvature terms. The conformal bienergy is conformally invariant in dimension four and its precise structure is motivated by the Paneitz operator from conformal geometry. The critical points of the conformal bienergy are called conformal biharmonic maps. Besides establishing a number of basic results on conformal biharmonic maps, we pay special attention to conformal biharmonic hypersurfaces in space forms. For hypersurfaces in spheres, we determine all conformal biharmonic hyperspheres and then we classify all conformal biharmonic generalized Clifford tori. Moreover, in sharp contrast to biharmonic hypersurfaces, we show that there also exist conformal biharmonic hypersurfaces of hyperbolic space, pointing out a fundamental difference between biharmonic and conformal biharmonic hypersurfaces. Finally, we also study the stability of the conformal biharmonic hyperspheres in spheres and explicitly compute their index and nullity. In particular, we obtain that the index of the equator $\mathbb{S}^4$ of $\mathbb{S}^5$ is zero, i.e., it is stable, while the index of the equator $\mathbb{S}^5$ of $\mathbb{S}^6$ is seven.

math.DG↗

arXiv2026

On the conformal-biharmonic stability of the identity map of Einstein manifolds

The identity map of an Einstein manifold is a critical point of both the classical energy functional and the conformal-bienergy functional. In this paper, we investigate the conformal-biharmonic stability of the identity map of compact Einstein manifolds of dimension at least four and with nonnegative scalar curvature, and we compare it with the harmonic stability, when the identity map is considered as a harmonic map. Somewhat surprisingly, we show that the conformal-biharmonic index coincides with the harmonic index, with a single notable exception: the four-dimensional Euclidean sphere. In this case, the identity map is unstable with respect to the energy functional, as shown independently by Mazet and Smith, whereas it is stable with respect to the conformal-bienergy functional.

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↗

arXiv2023

The energy density of biharmonic quadratic maps between spheres

In this paper, we first prove that a quadratic form from $\mathbb{S}^m$ to $\mathbb{S}^n$ is non-harmonic biharmonic if and only if it has constant energy density $(m+1)/2$. Then, we give a positive answer to an open problem concerning the structure of non-harmonic biharmonic quadratic forms. As a direct application, using classification results for harmonic quadratic forms, we infer classification results for non-harmonic biharmonic quadratic forms.

math.DG↗

arXiv2023

Segre embedding and biharmonicity

We consider the Segre embedding of the product $\mathbb{C}P^p\times\mathbb{C}P^q$ into $\mathbb{C}P^{p+q+pq}$ and study the biharmonicity of $M^p\times\mathbb{C}P^q$ and $M^p_1\times M^q_2$ as submanifolds of $\mathbb{C}P^{p+q+pq}$, where $M$ and $M_1$ are Lagrangian submanifolds of $\mathbb{C}P^p$ and $M_2$ is a Lagrangian submanifold of $\mathbb{C}P^q$. We find two new large classes of biharmonic submanifolds in complex projective space forms.

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↗

arXiv2023

On the biharmonic hypersurfaces with three distinct principal curvatures in space forms

In [16] there was proved that any biharmonic hypersurface with at most three distinct principal curvatures in space forms has constant mean curvature. At the very last step of the proof, the argument relied on the fact that the resultant of two polynomials is a non-zero polynomial. In this paper we point out that, in fact, there is a case, and only one, when this resultant is the zero polynomial and therefore the original proof is not fully complete. Further, we prove that in this special case we still obtain that the hypersurface has constant mean curvature.

math.DG↗

arXiv2022

Biconservative surfaces in the $4$-dimensional Euclidean sphere

In this paper, we study biconservative surfaces with parallel normalized mean curvature vector field ($PNMC$) in the $4$-dimensional unit Euclidean sphere $\mathbb{S}^4$. First, we study the existence and uniqueness of such surfaces. We obtain that there exists a $2$-parameter family of non-isometric abstract surfaces that admit a (unique) $PNMC$ biconservative immersion in $\mathbb{S}^4$. Then, we obtain the local parametrization of these surfaces in the $5$-dimensional Euclidean space $\mathbb{E}^5$.

math.DG↗

arXiv2022

Biharmonic homogeneous polynomial maps between spheres

In this paper we first prove a characterization formula for biharmonic maps in Euclidean spheres and, as an application, we construct a family of biharmonic maps from a flat $2$-dimensional torus $\mathbb{T}$ into the $3$-dimensional unit Euclidean sphere $\mathbb{S}^3$. Then, for the special case of maps between spheres whose components are given by homogeneous polynomials of the same degree, we find a more specific form for their bitension field. Further, we apply this formula to the case when the degree is $2$, and we obtain the classification of all proper biharmonic quadratic forms from $\mathbb{S}^1$ to $\mathbb{S}^n$, $n \geq 2$, from $\mathbb{S}^m$ to $\mathbb{S}^2$, $m \geq 2$, and from $\mathbb{S}^m$ to $\mathbb{S}^3$, $m \geq 2$.

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

PMC Biconservative Surfaces in Complex Space Forms

In this article we consider PMC surfaces in complex space forms, and we study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in non-flat complex space forms and we prove that they are biconservative if and only if totally real. Then, we find a Simons type formula for a well-chosen vector field constructed from the mean curvature vector field. Next, we prove a rigidity result for CMC biconservative surfaces in 2-dimensional complex space forms. We prove then a reduction codimension result for PMC biconservative surfaces in non-flat complex space forms. We conclude by constructing from the Segre embedding examples of CMC non-PMC biconservative submanifolds, and we also discuss when they are proper-biharmonic.

math.DG↗

arXiv2021

Biharmonic and biconservative hypersurfaces in space forms

We present some general properties of biharmonic and biconservative submanifolds and then survey recent results on such hypersurfaces in space forms. We also propose an alternative version for a well-known result of Nomizu and Smyth for hypersurfaces by replacing the CMC hypothesis with the more general condition of biconservativity.

math.DG↗

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 uniqueness of complete biconservative surfaces in $3$-dimensional space forms

Biconservative surfaces are surfaces with divergence-free stress-bienergy tensor. Simply connected, complete, non-$CMC$ biconservative surfaces in $3$-dimensional space forms were constructed working in extrinsic and intrinsic ways. Then, one raises the question of the uniqueness of such surfaces. In this paper we give a positive answer to this question.

math.DG↗

arXiv2020

Unique Continuation Property for Biharmonic Hypersurfaces in Spheres

We study properties of non-minimal biharmonic hypersurfaces of spheres. The main result is a CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres. We then deduce new rigidity theorems to support the Conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.

math.DG↗

arXiv2019

Complete biconservative surfaces in the hyperbolic space $\mathbb{H}^3$

We construct simply connected, complete, non-$CMC$ biconservative surfaces in the $3$-dimensional hyperbolic space $\mathbb{H}^3$ in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set of points where the gradient of the mean curvature function does not vanish is dense and has two connected components. In the intrinsic approach, we first construct a simply connected, complete abstract surface and then prove that it admits a unique biconservative immersion in $\mathbb{H}^3$. Working extrinsically, we use the images of the explicit parametric equations and a gluing process to obtain our surfaces. They are made up of circles (or hyperbolas, or parabolas, respectively) which lie in $2$-affine parallel planes and touch a certain curve in a totally geodesic hyperbolic surface $\mathbb{H}^2$ in $\mathbb{H}^3$.

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↗