arXiv subjects
Publications and source records attributed to P. Piccione.
Using an estimate on the number of critical points for a Morse-even function on the sphere $\mathbb S^m$, $m\ge1$, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to Euclidean balls. This yields also a multiplicity result for brake orbits in a potential well.
Brake orbits and homoclinics of autonomous dynamical systems correspond, via Maupertuis principle, to geodesics in Riemannian manifolds endowed with a metric which is singular on the boundary (Jacobi metric). Motivated by the classical, yet still intriguing in many aspects, problem of establishing multiplicity results for brake orbits and homoclinics, as done in [6, 7, 10], and by the development of a Morse theory in [8] for geodesics in such kind of metric, in this paper we study the related normal exponential map from a global perspective.
Let $(M,g)$ be a (complete) Riemannian surface, and let $Ω\subset M$ be an open subset whose closure is homeomorphic to a disk. We prove that if $\partialΩ$ is smooth and it satisfies a strong concavity assumption, then there are at least two distinct orthogonal geodesics in $\overlineΩ=Ω\bigcup\partialΩ$. Using the results given in [6], we then obtain a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. In our proof we shall use recent deformation results proved in [7].
Motivated by the use of degenerate Jacobi metrics for the study of brake orbits and homoclinics, we develop a Morse theory for geodesics in conformal metrics having conformal factors vanishing on a regular hypersurface of a Riemannian manifold.
We use a geometric construction to exhibit examples of autonomous Lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the same value of the maximum point. Similarly, we show examples of Hamiltonian systems that admit exactly two brake orbits in an annular potential region connecting the two connected components of the boundary of the potential well. These examples show that the estimates proven in [R. Giambò, F. Giannoni, P. Piccione, Arch. Ration. Mech. Anal. 200, (2011) 691-724] are sharp.
In this paper we prove the existence of real-analytic natural Hamiltonian systems - i.e. where H(q,p)=T(q,p)+V(q) in the 2N-dimensional real space, where N is any integer greater than 1 - with non critical energy levels E for the potential V such that the sublevel E of V is homeomorphic to the N-dimensional disk, and that only one brake orbit of energy E exists. A famous conjecture formulated by H. Seifert in 1948 claimed the existence of at least N distinct brake orbits for this situation.
We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds.
We prove that in conformal classes of metrics near the class of an Einstein metric (other than the standard round metric on a sphere) the Yamabe problem has a unique solution up to scaling. This is a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata.
We give estimates on the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of the total space of a Riemannian submersion.
In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for a class of Hamiltonian systems at a fixed energy level.
Following the lines of the celebrated Riemannian result of Gromoll and Meyer, we use infinite dimensional equivariant Morse theory to establish the existence of infinitely many geometrically distinct closed geodesics in a class of globally hyperbolic stationary Lorentzian manifolds.
Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.
We use the notion of generalized signatures at a singularity of a smooth curve of symmetric bilinear forms to determine a formula for the computation of the Maslov index in the case of a real-analytic path having possibly non transversal intersections. We discuss some applications of the theory, with special emphasis on the study of the Jacobi equation along a semi-Riemannian geodesic. The research work exposed in this paper originated from a suggestion given by the third author relating the spectral flow with the partial signatures. He pointed out several references and formulated the statement that in the invertible endpoints case, the spectral flow of a real analytic path of self-adjoint Fredholm operators is given by the sum of the odd partial signatures at each degeneracy instant. In the present version of the article, this statement is part of Proposition 2.9; totally, the contribution of the third author to the theory consists in Definition 2.3, formulas (2.1) and (2.2), the first statement of Proposition 2.4, parts of Remark 2.5, Definition 2.6, and parts of the statement and parts of the proof of Proposition 2.9. Apart from this original contribution, the material contained in this paper was entirely developed and written by the first two authors at the Universita' di Camerino (Italy), Universidade de Sao Paulo (Brazil).
We prove an extension of the Index Theorem for Morse-Sturm systems of the form $-V''+RV=0$, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M,g), we consider a geodesic $γ$ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, satisfied, for instance, if (M,g) is stationary, the theorem gives an equality between the index of the second variation of the action functional f at $γ$ and the sum of the {\em Maslov index} of $γ$ with the index of the metric g on P. Under generic circumstances, the Maslov index of $γ$ is given by an algebraic count of the P-focal points along $γ$. Using the Maslov index, we obtain the global Morse relations for geodesics between two fixed points in a stationary Lorentzian manifold.
The partial group algebra of a group G over a field K, denoted by K_{par}(G), is the algebra whose representations correspond to the partial representations of G over K-vector spaces. In this paper we study the structure of the partial group algebra K_{par}(G), where G is a finite group. In particular, given two finite abelian groups G_1 and G_2, we prove that if the characteristic of K is zero, then K_{par}(G_1) is isomorphic to K_{par}(G_2) if and only if G_1 is isomorphic to G_2.