The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713-756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C¹-Hamiltonian systems on the cotangent bundle of a C³-smooth compact manifold M without boundary, of a time 1-periodic C²-smooth Hamiltonian H:R×T^*M→R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H(−t,q,−p)=H(t,q,p) for any (t,q,p)∈R×T^*M, it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T^*M. If M is C⁵-smooth and dimM>1, H is of C⁴ class and independent of time t, then for any τ>0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H, L:R×TM→R, which is proved to be strongly convex and to have quadratic growth in the velocities yet.     

Subsheaves of the cotangent bundle

For any smooth projective variety, we study a birational invariant, defined by Campana which depends on the Kodaira dimension of the subsheaves of the cotangent bundle of the variety and its exterior powers. We provide new bounds for a related invariant in any dimension and in particular we show that it is equal to the Kodaira dimension of the variety, in dimension up to 4, if this is not negative.     

Differential invariants of the cotangent bundle

Vilnius Pedagigic Institute. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 30, No. 3, pp. 536–547, July–September, 1990.     

The cohomology of the cotangent bundle of Heisenberg groups

Given a parabolic subalgebra g₁×n of a semisimple Lie algebra, Kostant (Ann. Math. 1963) and Griffiths (Acta Math. 1963) independently computed the g₁ invariants in the cohomology group of n with exterior adjoint coefficients. By a theorem of Bott (Ann. Math. 1957), this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms. In this paper we determine explicitly the full module structure, over the symplectic group, of the cohomology group of the Heisenberg Lie algebra with exterior adjoint coefficients. This is the cohomology of the cotangent bundle of the Heisenberg group.     

A cotangent bundle slice theorem

This article concerns cotangent-lifted Lie group actions; our goal is to find local and “semi-global” normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle [C.-M. Marle, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rendiconti del Seminario Matematico, Università e Politecnico, Torino 43 (2) (1985) 227-251] and Guillemin and Sternberg [V. Guillemin, S. Sternberg, A normal form for the moment map, in: S. Sternberg (Ed.), Differential Geometric Methods in Mathematical Physics, in: Mathematical Physics Studies, vol. 6, D. Reidel, 1984]. The result applies to all proper cotangent-lifted actions, around points with fully-isotropic momentum values.We also present a “tangent-level” commuting reduction result and use it to characterise the symplectic normal space of any cotangent-lifted action. In two special cases, we arrive at splittings of the symplectic normal space. One of these cases is when the configuration isotropy group is contained in the momentum isotropy group; in this case, our splitting generalises that given for free actions by Montgomery et al. [R. Montgomery, J.E. Marsden, T.S. Ratiu, Gauged Lie-Poisson structures, Cont. Math. AMS 128 (1984) 101-114]. The other case includes all relative equilibria of simple mechanical systems. In both of these special cases, the new splitting leads to a refinement of the so-called reconstruction equations or bundle equations [J.-P. Ortega, Symmetry, reduction, and stability in Hamiltonian systems, PhD thesis, University of California, Santa Cruz, 1998; J.-P. Ortega, T.S. Ratiu, A symplectic slice theorem, Lett. Math. Phys. 59 (1) (2002) 81-93; M. Roberts, C. Wulff, J.S.W. Lamb, Hamiltonian systems near relative equilibria, J. Differential Equations 179 (2) (2002) 562-604]. We also note cotangent-bundle-specific local normal forms for symplectic reduced spaces.     

Conley conjecture and local Floer homology

In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit, the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold, the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.     

The projectivised cotangent bundle of an algebraic surface

Partially supported by project PS 90-0069 (Ministerio de Educacion y Ciencia, Spain) and by project Geometry of algebraic varieties, European Science Programme, SC 1-0398-C(A).     

Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256 (9) (2009) 2967-3034]

In lines 8-11 of Lu (2009) [18, p. 2977] we wrote: “For integer m?3, if M is Cm-smooth and Cm−1-smooth L:R×TM→R satisfies the assumptions (L1)-(L3), then the functional Lτ is C²-smooth, bounded below, satisfies the Palais-Smale condition, and all critical points of it have finite Morse indexes and nullities (see [1, Prop. 4.1, 4.2] and [4])”. However, as proved in Abbondandolo and Schwarz (2009) [2] the claim that Lτ is C²-smooth is true if and only if for every (t,q) the function v?L(t,q,v) is a polynomial of degree at most 2. So the arguments in Lu (2009) [18] are only valid for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this note we shall correct our arguments in Lu (2009) [18] with a new splitting lemma obtained in Lu (2011) [20].     

Restriction of the cotangent bundle to elliptic curves and Hilbert-Kunz functions

We describe the possible restrictions of the cotangent bundle to an elliptic curve . We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R₊-primary ideal in the graded section ring .     

Geometry of the cotangent bundle with Sasakian metrics and its applications

The main aim of this paper is to study paraholomorpic Sasakian metric and Killing vector field with respect to the Sasakian metric in the cotangent bundle.     

The role of the cotangent bundle in resolving ideals of fat points in the plane

We study the connection between the generation of a fat point scheme supported at general points in and the behaviour of the cotangent bundle with respect to some rational curves particularly relevant for the scheme. We put forward two conjectures, giving examples and partial results in support of them.     

Symplectic capacity and the Weinstein conjecture in certain cotangent bundles and Stein manifolds

We use the method proposed by H. Hofer. and C. Viterbo in [18] to calculate the Hofer-Zehnder capacity and prove the Weinstein conjecture in certain cotangent bundles and Stein manifolds.Supported in parts by SFB237 and the Forschunginstitut für Mathematik, ETH-Zentrum, Switzerland.     

The natural functions on the cotangent bundle of higher order vector tangent bundles over fibered manifolds

For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T ^*(J ^(r,s,q)(Y, R ^1,1)₀)^*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T ^*(J ^(r,s) (Y, R)₀)^*→R for any Y as above.     

On the BKS pairing for Kähler quantizations of the cotangent bundle of a Lie group

A natural one-parameter family of Kähler quantizations of the cotangent bundle T^∗K of a compact Lie group K, taking into account the half-form correction, was studied in [C. Florentino, P. Matias, J. Mourão, J.P. Nunes, Geometric quantization, complex structures and the coherent state transform, J. Funct. Anal. 221 (2005) 303-322]. In the present paper, it is shown that the associated Blattner-Kostant-Sternberg (BKS) pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work, from the point of view of [S. Axelrod, S. Della Pietra, E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Differential Geom. 33 (1991) 787-902]. The BKS pairing map is a composition of (unitary) coherent state transforms of K, introduced in [B.C. Hall, The Segal-Bargmann coherent state transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103-151]. Continuity of the Hermitian structure on the quantum bundle, in the limit when one of the Kähler polarizations degenerates to the vertical real polarization, leads to the unitarity of the corresponding BKS pairing map. This is in agreement with the unitarity up to scaling (with respect to a rescaled inner product) of this pairing map, established by Hall.     

The external multiplication for the Conley index

We construct an additional operation of the external multiplication on the cohomological Conley index defined by Mrozek for discrete semidynamical systems. The construction is based on the notion of the Conley index over a phase space introduced by Szybowski. We show how to apply the external multiplication to solve the problem of continuation of two isolated invariant sets and illustrate it by an example.     

Singular cotangent bundle reduction & spin Calogero–Moser systems

We develop a bundle picture for singular symplectic quotients of cotangent bundles acted upon by cotangent lifted actions for the case that the configuration manifold is of single orbit type. Furthermore, we give a formula for the reduced symplectic form in this setting. As an application of this bundle picture we consider Calogero–Moser systems with spin associated to polar representations of compact Lie groups.     

The Conley index for discontinuous vector fields

In this paper, we define the Conley index for a region of discontinuity D of a piecewise C ^k discontinuous vector field Z on an n-dimensional compact Riemannian smooth orientable manifold and prove it to be a homotopy invariant. This invariance is obtained by regularization of the discontinuous vector field. We use an adapted form of Lyapunov graph continuation to produce, in a few examples, a regularization of the discontinuous vector field with the property that the dynamics in a regularized neighborhood of D has the same Conley index as .