Group Of Hamiltonian Diffeomorphisms Research Articles

The unbounded Lagrangian spectral norm and wrapped Floer cohomology

We investigate the question of whether the spectral metric on the orbit space of a fiber in the disk cotangent bundle of a closed manifold, under the action of the compactly supported Hamiltonian diffeomorphism group, is bounded. We utilize wrapped Floer cohomology to define the spectral invariant of an admissible Lagrangian submanifold within a Weinstein domain. We show that the pseudo-metric derived from this spectral invariant is a valid Ham-invariant metric. Furthermore, we establish that the spectral metric on the orbit space of an admissible Lagrangian is bounded if and only if the wrapped Floer cohomology vanishes. Consequently, we prove that the Lagrangian Hofer diameter of the orbit space for any fiber in the disk cotangent bundle of a closed manifold is infinite.

Hofer geometry via toric degeneration

AbstractThe main theme of this paper is to use toric degeneration to study Hofer geometry by producing distinct homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We focus on the (complex n-dimensional) quadric hypersurface and the del Pezzo surfaces, and study two classes of distinguished Lagrangian submanifolds that appear naturally in a toric degeneration, namely the Lagrangian torus which is the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres that appear as vanishing cycles. For the quadrics, we prove that the group of Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and derive some superheaviness results. Along the way, we show that the toric degeneration is compatible with the Biran decomposition. This implies that for $$n=2$$ n = 2 , the Lagrangian fiber torus (Gelfand–Zeitlin torus) is Hamiltonian isotopic to the Chekanov torus, which answers a question of Y. Kim. We prove analogous results for the del Pezzo surfaces. We also discuss applications to $$C^0$$ C 0 symplectic topology.

Weighted nonlinear flag manifolds as coadjoint orbits

Abstract A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold endowed with a volume density. We study the geometry of Fréchet manifolds of weighted nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians. When the ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic nonlinear flags.

A Teichmüller space for negatively curved surfaces

AbstractWe first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally defined flat connection whose holonomy lies in the group of Hamiltonian diffeomorphisms of . Consideration of the holonomy necessitates an extension from Riemannian to Finsler metrics. The second part of the paper follows the Higgs bundle approach to flat connections adapted to this infinite‐dimensional group and focuses on a family of metrics, relying on a construction of O. Biquard, which is parametrised by the infinite‐dimensional space of CR functions on the unit circle bundle of a hyperbolic surface. This generates an alternative approach to defining a connection and offers the possibility of this vector space representing a moduli space which generalizes and includes the classical Teichmüller space.

Asymptotic representations of Hamiltonian diffeomorphisms and quantization

We show that for a special class of geometric quantizations with “small” quantum errors, the quantum classical correspondence gives rise to an asymptotic projective unitary representation of the group of Hamiltonian diffeomorphisms. As an application, we get an obstruction to Hamiltonian actions of finitely presented groups.

On the rank of π1(Ham)

We show that for any positive integer $k$ there exists a closed symplectic $4$-manifold, such that the rank of the fundamental group of the group of Hamiltonian diffeomorphisms is at least $k.$

Homogeneous quasimorphisms, $C^0$-topology and Lagrangian intersection

We construct an example of a non-trivial homogeneous quasimorphism on the group of Hamiltonian diffeomorphisms of the two and four dimensional quadric hypersurfaces which is continuous with respect to both the $C^0$-metric and the Hofer metric. This answers a variant of a question of Entov–Polterovich–Py which is one of the open problems listed in the monograph of McDuff–Salamon. Throughout the proof, we make extensive use of the idea of working with different coefficient fields in quantum cohomology rings. As a by-product of the arguments in the paper, we answer a question of Polterovich–Wu regarding homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms of the complex projective plane and prove some intersection results about Lagrangians in the four dimensional quadric hypersurface.

A Note on Partial Quasi-Morphisms and Products in Lagrangian Floer Homology in Cotangent Bundles

We define partial quasi-morphisms on the group of Hamiltonian diffeomorphisms of the cotangent bundle using the spectral invariants in Lagrangian Floer homology with conormal boundary conditions, where the product compatible with the PSS isomorphism and the homological intersection product is lacking.

Local rigidity, symplectic homeomorphisms, and coisotropic submanifolds

We introduce the notion of a point on a locally closed subset of a symplectic manifold being ‘locally rigid’ with respect to that subset, prove that this notion is invariant under symplectic homeomorphisms, and show that coisotropic submanifolds are distinguished among all smooth submanifolds by the property that all of their points are locally rigid. This yields a simplified proof of the Humilière–Leclercq–Seyfaddini theorem on the C 0 $C^0$ -rigidity of coisotropic submanifolds. Connections are also made to the ‘rigid locus’ that has previously been used in the study of Chekanov–Hofer pseudo-metrics on orbits of closed subsets under the Hamiltonian diffeomorphism group.

Coarse-graining and symplectic non-squeezing

We address aspects of coarse-graining in classical Statistical Physics from the viewpoint of the symplectic non-squeezing theorem. We make some comments regarding the implications of the symplectic non-squeezing theorem for the BBGKY hierarchy. We also see the cubic cells appearing in coarse-graining as a direct consequence of the uniqueness of Hofer’s metric on the group of Hamiltonian diffeomorphisms of the phase space.

On C0-Continuity of the Spectral Norm for Symplectically Non-Aspherical Manifolds

Abstract The purpose of this paper is to study the relation between the $C^0$-topology and the topology induced by the spectral norm on the group of Hamiltonian diffeomorphisms of a closed symplectic manifold. Following the approach of Buhovsky–Humilière–Seyfaddini, we prove the $C^0$-continuity of the spectral norm for complex projective spaces and negative monotone symplectic manifolds. The case of complex projective spaces provides an alternative approach to the $C^0$-continuity of the spectral norm proven by Shelukhin. We also prove a partial $C^0$-continuity of the spectral norm for rational symplectic manifolds. Some applications such as the Arnold conjecture in the context of $C^0$-symplectic topology are also discussed.

The action spectrum and $$C^0$$ symplectic topology

Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms. Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point, shows that the conjecture does not admit a direct generalization to the C^0 setting. However, in this paper we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.

Nonlinear flag manifolds as coadjoint orbits

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application, we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.

Critical symplectic connections on surfaces

The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphisms on the space of compatible almost complex structures that motivates study of extremal K\"ahler metrics. In particular, moment constant connections are critical, where a symplectic connection is \textit{critical} if it is critical, with respect to arbitrary variations, for the $L^{2}$-norm of the Cahen-Gutt moment map. This occurs if and only if the Hamiltonian vector field generated by its moment map image is an infinitesimal automorphism of the symplectic connection. This paper develops the study of moment constant and critical symplectic connections, following, to the extent possible, the analogy with the similar, but different, setting of constant scalar curvature and extremal K\"ahler metrics. It focuses on the special context of critical symplectic connections on surfaces, for which general structural results are obtained, although some results about the higher-dimensional case are included as well. For surfaces, projectively flat and preferred symplectic connections are critical, and the relations between these and other related conditions are examined in detail. The relation between the Cahen-Gutt moment map and the Goldman moment map for projective structures is explained.

Isotropic submanifolds and coadjoint orbits of the Hamiltonian group

We describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega)$ by implementing symplectic reduction for the dual pair associated to the Hamiltonian description of ideal fluids. The description is given in terms of nonlinear Grassmannians (manifolds of submanifolds) with additional geometric structures. Reduction at zero momentum yields the identification of coadjoint orbits with Grassmannians of isotropic volume submanifolds, slightly generalizing the results in Weinstein [1990] and Lee [2009]. At the other extreme, the case of a nondegenerate momentum recovers the identification of connected components of the nonlinear symplectic Grassmannian with coadjoint orbits, thereby recovering the result of Haller and Vizman [2004]. We also comment on the intermediate cases which correspond to new classes of coadjoint orbits. The description of these coadjoint orbits as well as their orbit symplectic form is obtained in a systematic way by exploiting the general properties of dual pairs of momentum maps. We also show that whenever the symplectic manifold $(M,\omega)$ is prequantizable, the coadjoint orbits that consist of isotropic submanifolds with total volume $a\in\mathbb{Z}$ are prequantizable. The prequantum bundle is constructed explicitly and, in the Lagrangian case, recovers the Berry bundle constructed in Weinstein [1990].

Integrability of central extensions of the Poisson Lie algebra via prequantization

We present a geometric construction of central S 1-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this to find nontrivial central S 1-extensions of the universal cover of the group of Hamiltonian diffeomorphisms. In the process, we obtain central S1-extensions of Lie groups that act by exact strict contact transformations.

Hamiltonian loops on the symplectic one-point blow up

We lift a Hamiltonian loop on a symplectic manifold to a Hamiltonian loop on the symplectic one-point blow up of a symplectic manifold. Then we use Weinstein's morphism to show that the lifted Hamiltonian loop has infinite order on the fundamental group of the group of Hamiltonian diffeomorphisms of the blown up manifold.

On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct explicit quasimorphisms on [Formula: see text] some of them are [Formula: see text]-continuous and vanish on all autonomous diffeomorphisms, and some of them are Calabi.

A Quasi-isometric embedding into the group of Hamiltonian diffeomorphisms with Hofer’s metric

We construct an embedding Φ of [0, 1]∞ into Ham(M, ω), the group of Hamiltonian diffeomorphisms of a suitable closed symplectic manifold (M, ω). We then prove that Φ is in fact a quasi-isometry. After imposing further assumptions on (M, ω), we adapt our methods to construct a similar embedding of ℝ ⊕ [0, 1]∞ into either Ham(M, ω) or , the universal cover of Ham(M, ω). Along the way, we prove results related to the filtered Floer chain complexes of radially symmetric Hamiltonians. Our proofs rely heavily on a continuity result for barcodes (as presented in [28]) associated to filtered Floer homology viewed as a persistence module.

Calabi quasimorphisms for monotone coadjoint orbits

We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group with Kostant–Kirillov–Souriau form. We show that this result follows from positivity results of Gromov–Witten invariants and the fact that the quantum product of Schubert classes is never zero.