Multiple closed geodesics on 3-spheres

This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.     

Orthogonal trajectories on Riemannian manifolds and applications to Plane Wave type spacetimes

We present a result on trajectories of a Lagrangian system joining two given submanifolds of a Riemannian manifold, under the action of an unbounded potential. As an application, we consider geodesics in a class of semi-Riemannian manifolds, the Plane Wave type spacetimes.     

Random sorting networks

A sorting network is a shortest path from 12?n to n?21 in the Cayley graph of Sn generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n→∞ the space-time process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Hölder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux.     

Closed geodesics on positively curved Finsler spheres

In this paper, we prove that for every Finsler n-sphere (Sn,F) for n?3 with reversibility λ and flag curvature K satisfying , either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp(πiμ) with an irrational μ. Furthermore, there always exist three prime closed geodesics on any (S³,F) satisfying the above pinching condition.     

A generalized distance and enhanced Ekeland’s variational principle for vector functions

We propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation. This result is shown to contain probably a large part of existing versions of Ekeland’s variational principle (EVP). We introduce the notion of a weak τ-function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced EVP for various settings, under relaxed lower semicontinuity assumptions. These principles conclude the existence not only of p-strict minimizers of p-perturbations of the considered vector function, but also p-sharp and p-strong minimizers. Our results are proved to be stronger than the classical EVP and many generalizations in the literature, even in the usual one-dimensional case, by numerous corollaries and examples. We include equivalent formulations of our enhanced EVP as well.     

The inertia of unicyclic graphs and the implications for closed-shells

The inertia of a graph is an integer triple specifying the number of negative, zero, and positive eigenvalues of the adjacency matrix of the graph. A unicyclic graph is a simple connected graph with an equal number of vertices and edges. This paper characterizes the inertia of a unicyclic graph in terms of maximum matchings and gives a linear-time algorithm for computing it. Chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is (properly) closed-shell, having exactly half of its eigenvalues greater than zero, because this designates a stable electron configuration. The inertia determines whether a graph is closed-shell, and hence the reported result gives a linear-time algorithm for determining this for unicyclic graphs.     

Extremal maps in Sobolev type inequalities: Some remarks

In this work we make some observations on the existence of extremal maps for sharp L²-Riemannian Sobolev type inequalities as Nash and logarithmic Sobolev ones. Among other results, we prove also that there exist smooth compact Riemannian manifolds with scalar curvature changing signal on which there exist extremal maps.     

Infinitely Peano differentiable functions in polynomially bounded o-minimal structures

Let R be an o-minimal expansion of a real closed field. We show that the definable infinitely Peano differentiable functions are smooth if and only if R is polynomially bounded.     

Harmonic sections of Riemannian vector bundles, and metrics of Cheeger-Gromoll type

We study harmonic sections of a Riemannian vector bundle E→M when E is equipped with a 2-parameter family of metrics hp,q which includes both the Sasaki and Cheeger-Gromoll metrics. For every k>0 there exists a unique p such that the harmonic sections of the radius-k sphere subbundle are harmonic sections of E with respect to hp,q for all q. In both compact and non-compact cases, Bernstein regions of the (p,q)-plane are identified, where the only harmonic sections of E with respect to hp,q are parallel. Examples are constructed of vector fields which are harmonic sections of E=TM in the case where M is compact and has non-zero Euler characteristic.     

Classification of homogeneous almost cosymplectic three-manifolds

The purpose of this paper is to classify all simply connected homogeneous almost cosymplectic three-manifolds. We show that each such three-manifold is either a Lie group G equipped with a left invariant almost cosymplectic structure or a Riemannian product of type R×N, where N is a Kähler surface of constant curvature. Moreover, we find that the Reeb vector field of any homogeneous almost cosymplectic three-manifold, except one case, defines a harmonic map.     

Dégénérescence de séries d’Eisenstein hyperboliques

In this work, we deal with hyperbolic Eisenstein series and more particularly, given a degenerating family S _l (l ≥ 0) of Riemann Surfaces with their canonical hyperbolic metrics, we work out the degeneration of hyperbolic Eisenstein series associated to the pinching geodesics in S _l . Our principal results are theorems 2.2, 4.1 et 4.2.     

Lightlike rays in stationary spacetimes with critical growth

The aim of this paper is to extend some previous results on the existence of lightlike geodesics joining a point to a line to the case of stationary Lorentzian manifolds whose metric coefficients have an optimal growth.     

Geodesic rays,Busemann functions and monotone twist maps

We study the action-minimizing half-orbits of an area-preserving monotone twist map of an annulus. We show that these so-called rays are always asymptotic to action-minimizing orbits. In the spirit of Aubry-Mather theory which analyses the set of action-minimizing orbits we investigate existence and properties of rays. By analogy with the geometry of the geodesics on a Riemannian 2-torus we define a Busemann function for every ray. We use this concept to prove that the minimal average action A() is differentiable at irrational rotation numbers while it is generically non-differentiable at rational rotation numbers (cf. also [18]). As an application of our results in the geometric framework we prove that a Riemannian 2-torus which has the same marked length spectrum as a flat 2-torus is actually isometric to this flat torus.     

Harmonic bundle solutions of topological-antitopological fusion in para-complex geometry

In this work we introduce the notion of a para-harmonic bundle, i.e. the generalization of a harmonic bundle [C.T. Simpson, Higgs-bundles and local systems, Inst. Hautes Etudes Sci. Publ. Math. 75 (1992) 5-95] to para-complex differential geometry. We show that para-harmonic bundles are solutions of the para-complex version of metric tt^∗-bundles introduced in [L. Schäfer, tt^∗-bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Differential Geom. Appl. 24 (1) (2006) 60-89]. Further we analyze the correspondence between metric para-tt^∗-bundles of rank 2r over a para-complex manifold M and para-pluriharmonic maps from M into the pseudo-Riemannian symmetric space GL(r,R)/O(p,q), which was shown in [L. Schäfer, tt^∗-bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Differential Geom. Appl. 24 (1) (2006) 60-89], in the case of a para-harmonic bundle. It is proven, that for para-harmonic bundles the associated para-pluriharmonic maps take values in the totally geodesic subspace GL(r,C)/Uπ(Cr) of GL(2r,R)/O(r,r). This defines a map Φ from para-harmonic bundles over M to para-pluriharmonic maps from M to GL(r,C)/Uπ(Cr). The image of Φ is also characterized in the paper.     

Boundedness for sublinear Duffing equations with time-dependent potentials

In this paper, we prove a sufficient and necessary condition for the boundedness of all solutions for the sublinear equation , where 0<α<1, p(t) and e(t) are smooth 1-periodic functions.     

Dinamica relativa di una particella con spin: un teorema di unicità

We prove anunicity theorem for spinning particles in the Papapetrou-Tulczyjew scheme; the theorem modifies the conclusive remarks of a preceding paper [1], i.e. the erroneous incompatibility of such scheme. The unicity is given using an intrinsic frame of reference (parallel to the generalized impulsion) and theresolvent equation of the differential problem, in terms of velocity (vectorial parameter).

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Sunto Si dimostra unteorema di unicità per lo schema di Papapetrou-Tulczyjew relativo ad una particella con spin in Relatività generale; il teorema vale a rettificare un precedente lavoro [1] che avanzava erroneamente, per tale schema, l’ipotesi di incompletezza. Il risultato si consegue operando (intrinsecamente) nelriferimento collineare all’impulso generalizzato, attraverso l’equazione risolvente per la velocità.

     

On harmonic tori in compact rank one symmetric spaces

In this paper we prove that, in contrast with the Sn and CPn cases, there are harmonic 2-tori into the quaternionic projective space HPn which are neither of finite type nor of finite uniton number; we also prove that any harmonic 2-torus in a compact Riemannian symmetric space which can be obtained via the twistor construction is of finite type if and only it is constant; in particular, we conclude that any harmonic 2-torus in CPn or Sn which is simultaneously of finite type and of finite uniton number must be constant.     

Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres

We prove that for every Q-homological Finsler 3-sphere (M, F) with a bumpy and irreversible metric F, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics. Huagui Duan: Partially supported by NNSF and RFDP of MOE of China. Yiming Long: Partially supported by the 973 Program of MOST, Yangzi River Professorship, NNSF, MCME, RFDP, LPMC of MOE of China, S. S. Chern Foundation, and Nankai University.     

Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface Σ=H⁻¹(1). If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.