Symplectic Manifold Research Articles

Nonpositively Curved Surfaces are Loewner

We show that every closed nonpositively curved surface satisfies Loewner’s systolic inequality. The proof relies on a combination of the Gauss–Bonnet formula with an averaging argument using the invariance of the Liouville measure under the geodesic flow. This enables us to find a disk with large total curvature around its center yielding a large area.

Homotopy BV-algebras in Hermitian geometry

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative BV∞-algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles.

Compactness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

Compactness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

On Darboux theorems for geometric structures induced by closed forms

This work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe mechanical systems), as well as certain cases of multisymplectic manifolds, while extends the Darboux theorem in new ways to k-symplectic and k-cosymplectic manifolds (all these structures appear in the geometric formulation of first-order classical field theories). Moreover, we discuss the existence of Darboux theorems for classes of precosymplectic, k-presymplectic, k-precosymplectic, and premultisymplectic manifolds, which are the geometrical structures underlying some kinds of singular field theories, i.e. with locally non-invertible Legendre maps. Approaches to Darboux theorems based on flat connections associated with geometric structures are given, while new results on polarisations for (k-)(pre)(co)symplectic structures arise.

Existence of generating families on Lagrangian cobordisms

For an embedded exact Lagrangian cobordism between Legendrian submanifolds in the 1-jet bundle, we prove that a generating family linear at infinity on the Legendrian at the negative end extends to a generating family linear at infinity on the Lagrangian cobordism after stabilization if and only if the formal obstructions vanish. In particular, a Lagrangian filling with trivial stable Lagrangian Gauss map admits a generating family linear at infinity.

Homological Lagrangian monodromy for some monotone tori

Given a Lagrangian submanifold L in a symplectic manifold X , the homological Lagrangian monodromy group \mathcal{H}_{L} describes how Hamiltonian diffeomorphisms of X preserving L setwise act on H_{*}(L) . We begin a systematic study of this group when L is a monotone Lagrangian n -torus. Among other things, we describe \mathcal{H}_{L} completely when L is a monotone toric fibre, make significant progress towards classifying the groups that can occur for n=2 , and make a conjecture for general n . Our classification results rely crucially on the arithmetic properties of Floer cohomology rings.

Geometry of bundle-valued multisymplectic structures with Lie algebroids

We study multisymplectic structures taking values in vector bundles with connections from the viewpoint of the Hamiltonian symmetry. We introduce the notion of bundle-valued n-plectic structures and exhibit some properties of them. In addition, we define bundle-valued homotopy momentum sections for bundle-valued n-plectic manifolds with Lie algebroids to discuss momentum map theories in both cases of quaternionic Kähler manifolds and hyper-Kähler manifolds. Furthermore, we generalize the Marsden-Weinstein-Meyer reduction theorem for symplectic manifolds and construct two kinds of reductions of vector-valued 1-plectic manifolds.

Special Kähler geometry and holomorphic Lagrangian fibrations

Special Kähler geometry and holomorphic Lagrangian fibrations

Poisson geometric formulation of quantum mechanics

We study the Poisson geometrical formulation of quantum mechanics for finite dimensional mixed and pure states. Equivalently, we show that quantum mechanics can be understood in the language of classical mechanics. We review the symplectic structure of the Hilbert space and identify its canonical coordinates. We extend the geometric picture to the space of density matrices DN+. We find it is not symplectic but admits a linear su(N) Poisson structure. We identify Casimir surfaces of DN+ and show that the space of pure states PN≡CPN−1 is one of its symplectic submanifolds which is an intersection of primitive Casimirs. We identify generic symplectic submanifolds of DN+ and calculate their dimensions. We find that DN+ is singularly foliated by the symplectic leaves of varying dimensions, also known as coadjoint orbits. We also find an ascending chain of Poisson submanifolds DNM⊂DNM+1 for 1 ≤ M ≤ N − 1. Each such Poisson submanifold DNM is obtained by tracing out the CM states from the bipartite system CN×CM and is an intersection of N − M primitive Casimirs of DN+. Their Poisson structure is induced from the symplectic structure of the bipartite system. We also show their foliations. Finally, we study the positive semi-definite geometry of the symplectic submanifold ENM consisting of the mixed states with maximum entropy in DNM.

Random circular billiards on surfaces of constant curvature: Pseudo integrability and mixing

Given a random map [Formula: see text], we define a random billiard map on a surface of constant curvature (Euclidean plane, hyperbolic plane, or the sphere). The Liouville measure is invariant for this billiard map. Finally, we show some dynamical properties such as ergodicity in the case of random circular billiards.

Asymptotics of the sticky particles evolution

We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the sticky particles dynamics is confined to a compact set.

On the spaces of $(d + d^{c})$-harmonic forms and $(d + d^{\Lambda})$-harmonic forms on almost Hermitian manifolds and complex surfaces

In this paper, we study the spaces of (d+d^{c}) -harmonic forms and of (d+d^{\Lambda}) -harmonic forms, a natural generalization of the spaces of Bott–Chern harmonic forms (respectively, symplectic harmonic forms) from complex (respectively, symplectic) manifolds to almost Hermitian manifolds. We apply the same techniques to compact complex surfaces, computing their Bott–Chern and Aeppli numbers and their spaces of (d+d^{\Lambda}) -harmonic forms. We give several applications to compact quotients of Lie groups by a lattice.

Symplectic rigidity of O’Grady’s tenfolds

We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O’Grady’s 10 10 -dimensional deformation type is trivial.

Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence

Let Y be a prequantization bundle over a closed spherically monotone symplectic manifold Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}. Adapting an idea due to Diogo and Lisi, we study a split version of Rabinowitz Floer homology for Y in the following two settings. First, Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} is a symplectic hyperplane section of a closed symplectic manifold X satisfying a certain monotonicity condition; in this case, X\\Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X {{\\setminus }} \\Sigma $$\\end{document} is a Liouville filling of Y. Second, the minimal Chern number of Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} is greater than one, which is the case where the Rabinowitz Floer homology of the symplectization R×Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R} \ imes Y$$\\end{document} is defined. In both cases, we construct a Gysin-type exact sequence connecting the Rabinowitz Floer homology of X\\Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X{\\setminus }\\Sigma $$\\end{document} or R×Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R} \ imes Y$$\\end{document} and the quantum homology of Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}. As applications, we discuss the invertibility of a symplectic hyperplane section class in quantum homology, the isotopy problem for fibered Dehn twists, the orderability problem for prequantization bundles, and the existence of translated points. We also provide computational results based on the exact sequence that we construct.

On a theorem by Schlenk

In this paper we prove a generalisation of Schlenk’s theorem about the existence of contractible periodic Reeb orbits on stable, displaceable hypersurfaces in symplectically aspherical, geometrically bounded, symplectic manifolds, to a forcing result for contractible twisted periodic Reeb orbits. We make use of holomorphic curve techniques for a suitable generalisation of the Rabinowitz action functional in the stable case in order to prove the forcing result. As in Schlenk’s theorem, we derive a lower bound for the displacement energy of the displaceable hypersurface in terms of the action value of such periodic orbits. The main application is a forcing result for noncontractible periodic Reeb orbits on quotients of certain symmetric star-shaped hypersurfaces. In this case, the lower bound for the displacement energy is explicitly given by the difference of the two periods. This theorem can be applied to many physical systems including the Hénon–Heiles Hamiltonian and Stark–Zeeman systems. Further applications include a new proof of the well-known fact that the displacement energy is a relative symplectic capacity on R2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{2n}$$\\end{document} and that the Hofer metric is indeed a metric.

A Hermitian refinement of symplectic Clifford analysis

In this paper, we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure on the canonical symplectic manifold . This gives rise to two symplectic Dirac operators and (in the sense of Habermann), leading to a ‐invariant system of equations on . We discuss the solution space for this system, culminating in a Fischer decomposition for the space of (harmonic) polynomials on with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra.

Basic Remarks on Lagrangian Submanifolds of Hyperkähler Manifolds

Abstract This note presents basic restrictions on the topology of Lagrangian surfaces of hyper-Kähler $4$-folds and a remark on the interaction of a Lagrangian subvariety of a hyper-Kähler variety with a Lagrangian fibration of the latter.

The Arnold conjecture for singular symplectic manifolds

In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of bm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^m$$\\end{document}-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number of 1-periodic Hamiltonian orbits for b2m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^{2m}$$\\end{document}-symplectic manifolds depending only on the topology of the manifold. Moreover, for bm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^m$$\\end{document}-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions.

Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds

We consider higher generalizations of both a (twisted) Poisson structure and the equivariant condition of a momentum map on a symplectic manifold. On a Lie algebroid over a (pre-)symplectic and (pre-)multisymplectic manifold, we introduce a Lie algebroid differential form called a compatible $E$-$n$-form. This differential form satisfies a compatibility condition, which is consistent with both the Lie algebroid structure and the (pre-)(multi)symplectic structure. There are many interesting examples such as a Poisson structure, a twisted Poisson structure and a twisted $R$-Poisson structure for a pre-$n$-plectic manifold. Moreover, momentum maps and momentum sections on symplectic manifolds, homotopy momentum maps and homotopy momentum sections on multisymplectic manifolds have this structure.

Isolated hypersurface singularities, spectral invariants, and quantum cohomology

Abstract We study the relation between isolated hypersurface singularities (e.g., ADE) and the quantum cohomology ring by using spectral invariants, which are symplectic measurements coming from Floer theory. We prove, under the assumption that the quantum cohomology ring is semi-simple, that (1) if the smooth Fano variety degenerates to a Fano variety with an isolated hypersurface singularity, then the singularity has to be an A m {A_{m}} -singularity, (2) if the symplectic manifold contains an A m {A_{m}} -configuration of Lagrangian spheres, then there are consequences for the Hofer geometry, and that (3) the Dehn twist reduces spectral invariants.