arXiv subjects

Paolo Piccione

Publications and source records attributed to Paolo Piccione.

At least 19 recordsLinked to original sources

arXiv2026

Delaunay solutions to the fractional Hartree equation with critical growth

We study positive solutions of the critical fractional Hartree equation with a non-removable isolated singularity at the origin. This equation is doubly nonlocal, involving both the fractional Laplacian and a Riesz convolution potential. We first prove that every positive singular solution is radially symmetric about the origin, by combining the Caffarelli--Silvestre extension with the method of moving spheres. We then establish the existence of Delaunay-type periodic singular solutions. After the Emden--Fowler transformation, the Hartree convolution survives as a genuinely nonlocal integral term, so that the resulting periodic equation cannot be reduced to an ordinary differential equation. We construct nonconstant periodic solutions for all sufficiently large periods by minimizing a Rayleigh-type quotient in a periodic fractional Sobolev space.

math.AP↗

arXiv2026

Zoll manifolds with boundary

We introduce and study Zoll manifolds with boundary: compact Riemannian manifolds with smooth boundary such that every geodesic issuing orthogonally from the boundary returns orthogonally and is nowhere tangent to it. We first show that all such free boundary geodesics are embedded and have a common length, and that the boundary has at most two connected components. If there are two components, we prove that the manifold is a product of an interval with a closed manifold. When the boundary is connected, we show that the manifold is a tubular neighborhood of a closed embedded submanifold, the "soul", and that the complement of the soul is diffeomorphic to a half-open cylinder over the boundary. We further prove that all free boundary geodesics are maximally degenerate critical points of the energy functional and have the same Morse index, which equals the multiplicity of the unique focal point occurring at the midpoint of each geodesic. The projection from the boundary to the soul is then either a nontrivial two-fold covering or a smooth sphere bundle, according to the value of this index. As applications, we obtain a complete classification of Zoll surfaces with boundary and of three-dimensional Zoll manifolds with boundary.

math.DG↗

arXiv2026

Satellite Links with Multiple Full Twists and Single-Twist Companions

We study the relationship between the number of full twists in positive braid representations of satellite links and their companion links. We construct infinitely many satellite links that admit positive braid representations with arbitrarily many full twists, while their companion links do not admit any positive braid representation with more than one full twist. This exhibits an unexpected divergence between the braid-theoretic complexity of a satellite link and that of its companion.

math.GT↗

arXiv2025

Bifurcations of Clifford tori in ellipsoids

We prove that 3-dimensional ellipsoids invariant under a 2-torus action contain infinitely many distinct immersed minimal tori, with at most one exception. These minimal tori bifurcate from the 2-torus orbit of largest volume at a dense set of eccentricities, and remain invariant under a circle.

math.DG↗

arXiv2025

Nonplanar minimal spheres in ellipsoids of revolution

We use global bifurcation techniques to establish the existence of arbitrarily many geometrically distinct nonplanar embedded smooth minimal 2-spheres in sufficiently elongated 3-dimensional ellipsoids of revolution. More precisely, we quantify the growth rate of the number of such minimal spheres, and describe their asymptotic behavior as the ellipsoids converge to a cylinder.

math.DG↗

arXiv2025

A general nonuniqueness result for Yamabe-type problems for conformally variational Riemannian invariants

Given a conformally variational scalar Riemannian invariant $I$, we identify a sufficient condition for a compact Riemannian manifold to admit finite regular coverings with many nonhomothetic conformal rescalings with $I$ constant. We also identify a sufficient condition for the universal cover to admit infinitely many geometrically distinct periodic conformal rescalings with $I$ constant. Using these conditions, we improve known nonuniqueness results for the $Q$-curvatures of orders two, four, and six, and establish nonuniqueness results for higher-order $Q$-curvatures and renormalized volume coefficients.

math.DG↗

arXiv2025

Nonhomothetic complete periodic metrics with constant scalar curvature

We show that there are infinitely many pairwise nonhomothetic, complete, periodic metrics with constant scalar curvature that are conformal to the round metric on $S^n\setminus S^k$, where $k < \frac{n-2}{2}$. These metrics are obtained by pulling back Yamabe metrics defined on products of $S^{n-k-1}$ and compact hyperbolic $(k+1)$-manifolds. Our main result proves that these solutions are generically distinct up to homothety. The core of our argument relies on classical rigidity theorems due to Obata and Ferrand, which characterize the round sphere by its conformal group.

math.DG↗

arXiv2025

Stable invariant manifold for generalized ODEs with applications to measure differential equations

This paper establishes the stable invariant manifold for a new kind of differential equations defined by Kurzweil integral, so-called {\em generalized ODEs} on a Banach space. The nonlinear generalized ODEs are formulated as $$ \frac{dz}{dτ}=D[Λ(t)z+F(z,t)], $$ where $Λ(t)$ is a bounded linear operator on a Banach space $\mathscr{Z}$ and $F(z,t)$ is a nonlinear Kurzweil integrable function on $\mathscr{Z}$. The letter $D$ represents that generalized ODEs are defined via its solution, and $\frac{dz}{dτ}$ only a notation. Hence, generalized ODEs are fundamentally a notational representation of a class of integral equations. Due to the differences between the theory of generalized ODEs and ODEs, it is difficult to extended the stable manifold theorem of ODEs to generalized ODEs. In order to overcome the difficulty, we establish a generalized Lyapunov-Perron equation in the frame of Kurzweil integral theory. Subsequently, we present a stable invariant manifold theorem for nonlinear generalized ODEs when their linear parts exhibit an exponential dichotomy. As effective applications, we finally derive results concerning the existence of stable manifold for measure differential equations and impulsive differential equations.

math.CA↗

arXiv2025

Local asymptotics for singular solutions to critical Hartree equations

We investigate the qualitative properties of a critical Hartree equation defined on punctured domains. Our study has two main objectives: analyzing the asymptotic behavior near isolated singularities and establishing radial symmetry of positive singular solutions. First, employing asymptotic analysis, we characterize the local behavior of solutions near the singularity. Specifically, we show that, within a punctured ball, solutions behave like the blow-up limit profile. This is achieved through classification results for entire bubble solutions, a standard blow-up procedure, and a removable singularity theorem, yielding sharp upper and lower bounds near the origin. To run the blow-up analysis, we develop an asymptotic integral version of the moving spheres technique, a technique of independent interest. Second, we establish the radial symmetry of blow-up limit solutions using an integral moving spheres method. On the technical level, we apply the integral dual method from Jin, Li, Xiong \cite{MR3694645, arxiv:1901.01678} to provide local asymptotic estimates within the punctured ball and to prove that solutions in the entire punctured space are radially symmetric with respect to the origin. Our results extend seminal theorems of Caffarelli, Gidas, and Spruck \cite{MR982351} to the setting of Hartree equations.

math.AP↗

arXiv2025

Parallel differential forms of codegree two, and three-forms in dimension six

For a differential form on a manifold, having constant components in suitable local coordinates trivially implies being parallel relative to a torsion-free connection, and the converse implication is known to be true for $p$-forms in dimension $n$ when $p=0,1,2,n-1,n$. We prove the converse for $(n-2)$-forms, and for 3-forms when $n=6$, while pointing out that it fails to hold for Cartan 3-forms on all simple Lie groups of dimensions $n\ge8$ as well as for $(n,p)=(7,3)$ and $(n,p)=(8,4)$, where the 3-forms and 4-forms arise in compact simply connected Riemannian manifolds with exceptional holonomy groups. We also provide geometric characterizations of 3-forms in dimension six and $(n-2)$-forms in dimension $n$ having the constant-components property mentioned above, and describe examples illustrating the fact that various parts of these geometric characterizations are logically independent.

math.DG↗

arXiv2024

From bubbles to clusters: Multiple solutions to the Allen--Cahn system

We extend previous works on the multiplicity of solutions to the Allen-Cahn system on closed Riemannian manifolds by considering an arbitrary number of phases. Specifically, we show that on parallelizable manifolds, the number of solutions is bounded from below by topological invariants of the underlying manifold, provided the temperature parameter and volume constraint are sufficiently small. The Allen-Cahn system naturally arises in phase separation models, where solutions represent the distribution of distinct phases in a multi-component mixture. As the temperature parameter approaches zero, the system's energy approximates the multi-isoperimetric profile, leading to solutions concentrating in regions resembling isoperimetric clusters. For two or three phases, these results rely on classifying isoperimetric clusters. However, this classification remains incomplete for a larger number of phases. To address this technical issue, we employ a "volume-fixing variations" approach, enabling us to establish our results for any number of phases and small volume constraints. This offers deeper insights into phase separation phenomena on manifolds with arbitrary geometry.

math.AP↗

arXiv2024

Special Ricci-Hessian equations on Kähler manifolds

Special Ricci-Hessian equations on Kähler manifolds $(M,g)$, as defined by Maschler [Ann. Global Anal. Geom. 34 (2008), 367--380] involve functions $τ$ on $M$ and state that, for some function $α$ of the real variable $τ$, the sum of $α\nabla dτ$ and the Ricci tensor equals a functional multiple of the metric $g$, while $α\nabla dτ$ itself is assumed to be nonzero almost everywhere. Three well-known obvious ``standard'' cases are provided by (non-Einstein) gradient Kähler-Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler-Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $α=2\cotτ$, or $α=2\cothτ$, or $α=2\tanhτ$. We also use the Cartan-Kähler theorem to prove that these three types are actually realized in a ``nonstandard'' way.

math.DG↗

arXiv2024

A note on the regularity and the existence of Riemannian splines

In this paper, we present a comprehensive proof concerning the regularity of critical points for the spline energy functional on Riemannian manifolds, even for the general higher-order case. Although this result is widely acknowledged in the literature, a detailed proof was previously absent. Our proof relies on a generalization of the DuBois-Reymond Lemma. Furthermore, we establish the existence of minimizers for the spline energy functional in cases where multiple interpolation points are prescribed alongside just one velocity.

math.AP↗

arXiv2024

Nijenhuis geometry of parallel tensors

A tensor -- meaning here a tensor field $Θ$ of any type $(p,q)$ on a manifold -- may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of integrability for differential $q$-forms, $q=0,1,2,n-1,n$ (in dimension $n$), vectors, bivectors, symmetric $(2,0)$ and $(0,2)$ tensors, as well as complex-diagonalizable and nilpotent tensors of type $(1,1)$. In most cases, integrability is equivalent to algebraic constancy of $Θ$ coupled with the vanishing of one or more suitably defined Nijenhuis-type tensors, depending on $Θ$ via a quasilinear first-order differential operator. For $(p,q)=(1,1)$, they include the ordinary Nijenhuis tensor.

math.DG↗

arXiv2024

Multiplicity results for mass constrained Allen-Cahn equations on Riemannian manifolds with boundary

We present multiplicity results for mass constrained Allen-Cahn equations on a Riemannian manifold with boundary, considering both Neumann and Dirichlet conditions. These results hold under the assumptions of small mass constraint and small diffusion parameter. We obtain lower bounds on the number of solutions according to the Lusternik--Schnirelmann category of the manifold in case of Dirichlet boundary conditions and of its boundary in the case of Neumann boundary conditions. Under generic non-degeneracy assumptions on the solutions, we obtain stronger results based on Morse inequalities. Our approach combines topological and variational methods with tools from Geometric Measure Theory.

math.AP↗

arXiv2024

Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge

We consider an autonomous, indefinite Lagrangian admitting an infinitesimal symmetry whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point to a flow line of the infinitesimal generator $K$. By utilizing the invariance of the Lagrangian under the flow of $K$, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the ``arrival time'', seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When the Lagrangian is positively homogeneous of degree two in the velocities, the resulting equation establishes a variational principle that extends the Fermat's principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine.

math.DS↗

arXiv2023

Compact flat manifolds and reducibility

Hiss and Szczepański proved in 1991 that the holonomy group of any compact flat Riemannian manifold, of dimension at least two, acts reducibly on the rational span of the Euclidean lattice associated with the manifold via the first Bieberbach theorem. Geometrically, their result states that such a manifold must admit a nonzero proper parallel distribution with compact leaves. We study algebraic and geometric properties of the sublattice-spanned holonomy-invariant rational vector subspaces that exist due to the above theorem, and of the resulting compact-leaf foliations of compact flat manifolds. The class consisting of the former subspaces, in addition to being closed under spans and intersections, also turns out to admit (usually nonorthogonal) complements. As for the latter foliations, we provide descriptions, first -- and foremost -- of the intrinsic geometry of their generic leaves in terms of that of the original flat manifold and, secondly -- as an essentially obvious afterthought -- of the leaf-space orbifold. The general conclusions are then illustrated by examples in the form of generalized Klein bottles.

math.DG↗

arXiv2023

Nonuniqueness results for constant sixth order $Q$-curvature metrics on spheres with higher dimensional singularities

We prove nonuniqueness results for constant sixth order $Q$-metrics on complete locally conformally flat $n$-dimensional Riemannian manifolds with $n\geqslant 7$. More precisely, assuming a positive Green function exists for the sixth order GJMS operator, our objective is two-fold. First, we use a classical bifurcation technique to prove that there exists infinitely many constant $Q$-curvature metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$. As a by-product, we find the sixth order Yamabe invariant on this product manifold can be arbitrarily close to that of the round dimensional sphere, generalizing a result of Schoen about the classical Yamabe invariant. Second, when the underlying manifold is noncompact, we apply a bifurcation technique on Riemannian covering to construct infinitely many complete metrics with constant sixth order $Q$-curvature conformal to $\mathbb{S}^{n_1} \times \mathbb{R}^{n_2}$ or $\mathbb{S}^{n_1} \times \mathbb{H}^{n_2}$, where $n_1+n_2\geqslant 7$. Consequently, we obtain infinitely many solutions to the singular constant GJMS equation on round spheres $\mathbb{S}^n\setminus \mathbb{S}^k$ blowing up along a minimal equatorial subsphere with $0 \leqslant k<\frac{n-6}{2}$; this dimension restriction is sharp in the topological sense.

math.DG↗