We introduce here a natural functional associated to any $b \in QH_* (M, ω)$: \emph{spectral length functional}, on the space of "generalized paths" in $ \text {Ham}(M, ω)$, closely related to both the Hofer length functional and spectral invariants and establish some of its properties. This functional is smooth on its domain of definition, and moreover the nature of extremals of this functional suggests that it may be variationally complete, in the sense that any suitably generic element of $ \widetilde{\text {Ham}}(M, ω)$ is connected to $id$ by a generalized path minimizing spectral length. Rather strong evidence is given for this when $M=S ^{2}$, where we show that all the Lalonde-McDuff Hamiltonian symplectomorphisms are joined to id by such a path. We also prove that the associated norm on $ {\text {Ham}}(M, ω)$ is non-degenerate and bounded from below by the the spectral norm. If the spectral length functional is variationally complete the associated norm reduces to the spectral norm.
This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by the Floer homologies of powers of the symplectomorphism, the other algebraic invariants of symplectic field theory provide natural generalizations of symplectic Floer homology. For symplectically aspherical manifolds and Hamiltonian symplectomorphisms we study the moduli spaces o…
For a closed symplectic manifold $(M,ω)$ with compatible Riemannian metric $g$ we study the Sobolev $H^1$ geometry of the group of all $H^s$ diffeomorphisms on $M$ which preserve the symplectic structure. We show that, for sufficiently large $s$, the $H^1$ metric admits globally defined geodesics and the corresponding exponential map is a non-linear Fredholm map of index zero. Finally, we show that the $H^1$ metric carries conjugate points via some simple examples.
We prove that the vector space R^d of any finite dimension d with the standard metric embeds in a bi-Lipschitz way into the group of area-preserving diffeomorphisms G of the two-sphere endowed with the L^p-metric for p>2. Along the way we show that the L^p-metric on the group G is unbounded for p>2 by elementary methods.
In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into \CP^2 by using a new way to desingularize orbifold blow ups Z of the weighted projective space \CP^2_{1,m,n}. We now use a related method to construct symplectomorphisms of these spaces Z. This allows us to construct some well known Fano 3-folds (including the Mukai--Umemura 3-fold) in purely symplectic terms, using a classification by Tolman of a particular class of Hamiltonian S^1-manifolds. We also…
Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We show that for certain closed symplectic manifolds, including complex projective spaces and Grassmannians, the universal cover of the group of Hamiltonian symplectomorphisms of the manifold has infinite commutator length. In particular, we present explicit examples of elements in …
Let $P$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $X_{P}$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $t = \sqrt{-1} s$. We relate the quantization of $X_{P}$ in two different toric Kähler polarizations by taking the time-$\sqrt{-1} s$ Hamiltonian "flow" of strongly convex functions on the moment polytope $P$. By taking $s$ to infinity, we obtain the quantization of $X_{P}$ in the (singular) real toric polarization. Recall that $X_{P}$ has an open dens…
Each loop $ψ$ in the group $\text{Ham}(M)$ of Hamiltonian diffeomorphisms of a symplectic manifold $M$ determines a fibration $E$ on $S^2$, whose coupling class \cite{G-L-S} is denoted by $c$. If $VTE$ is the vertical tangent bundle of $E$, we relate the characteristic number $\int_E c_1(VTE)c^n$ with the Maslov index of the linearized flow $ψ_{t*}$ and the Chern class $c_1(TM)$. We give the value of this characteristic number for loops of Hamiltonian symplectomorphisms of Hirzebruch surfaces.
In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundle of $\mathbf{RP}^{2}$ and of $\mathbf{C}^*\times\mathbf{C}$. We also make progress in the case of the $A_n$-Milnor fibres: here we can show that the (compactly supported) Hamiltonian group is contractible and that the symplectic mapping class group embeds in the braid group on $…
By a theorem of Banyaga the group of diffeomorphisms of a manifold $P$ preserving a regular contact form $α$ is a central $S^1$ extension of the commutator of the group of symplectomorphisms of the base $B = P/S^1$. We show that if $T$ is a Hamiltonian maximal torus in the group of symplectomorphism of $B$, then its preimage under the extension map is a maximal torus not only in the group $\Diff(P, α)$ of diffeomorphisms of $P$ preserving $α$ but also in the much bigger group of contactomorphisms $\Diff (P, ξ)$, the group of diffeomorphism of $P$ preserving the contact distribution $ξ= \ker α$…
We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if $M$ is a simply connected symplectic 4-manifold with $b_{2}\geq 3$, and if $\widetilde{M}_δ$ denotes a blow-up of $M$ of small enough capacity $δ$, then the rational cohomology algebra of the Hamiltonian group of $\widetilde{M}_δ)$ is not finitely generated. Both results are based on the fact that…
Consider the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold $(M,\om)$ with the Hofer $L^{\infty}$-norm. A path in $\Ham^c(M)$ will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional $\Ll$. In this paper, we give a necessary condition for a path $\ga$ to be a geodesic. We also develop a necessary condition for a geodesic to be stable, that is, a local minimum for $\Ll$. This condition is related to the existence of periodic orbits for the linearization of the path, and so extends Ustilo…
This paper concerns the action of linear symplectomorphisms on linear symplectic forms by conjugation in even dimensions. We prove that pfaffian and $-\frac{1}{2}\operatorname{tr}(JA)$ (sum function) of $A$ are invariants on the action. We use these invariants to provide a complete description of the orbit space in dimension four. In addition, we investigate the geometric shapes of the individual orbits in dimension four. In symplectic geometry, our classification result in dimension four provides a necessary condition for two symplectic forms on $\mathbb{R}^{4}$ to be intertwined by symplecto…
This paper studies the question of when a loop $ϕ$ in the group Symp$(M,ω)$ of symplectomorphisms of a symplectic manifold $(M,ω)$ is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if $ϕ$ is Hamiltonian with respect to $ω$, and if $ϕ'$ is a small perturbation of $ϕ$ that preserves another symplectic form $ω'$, then $ϕ'$ is Hamiltonian with respect to $ω'$. This allows us to get some new information on the structure of the flux group,…
A closed 3-form $H \in Ω^3_0(M)$ defines an extension of $Γ(TM)$ by $Ω^2_0(M)$. This fact leads to the definition of the group of $H$-twisted Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold. The Hamiltonian reduction in the category of generalized complex geometry is then constructed. The definitions and constructions are natural extensions of the corresponding ones in the symplectic geometry. We describe cutting in generalized complex geometry to show that it's a general phenomenon in …
Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: Ω\text {Ham}(M, ω) \to \mathbb{R}$, with Gromov-Witten/Floer theory, for monotone symplectic manifolds $ (M, ω) $. This gives some immediate restrictions on the topology of the group of Hamiltonian symplectomorphisms (possibly relative to the Hofer length functional), and a criterion for non-existence of certain higher index geodesics for the Hofer length functional. The argument is based on a certain automatic transversality p…
We give a characterisation of Atiyah's and Hitchin's transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces.
In this article, we treat G_2-geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G_2-structure; in particular, we discuss existence and make a number of identifications of the spaces of Hamiltonian structures associated to the two multisymplectic structures associated to an integrable G_2-structure. Along the way, we prove some results in multisymplectic geometry that are generalizations of results from symplectic geometry.
This paper studies the geometry of the group of all co-Hamiltonian diffeomorphisms of a compact cosymplectic manifold $(M, ω, η)$. The fix-point theory for co-Hamiltonian diffeomorphisms is studied, and we use Arnold's conjecture to predict the exact minimum number of fix point that such a diffeomorphism must have (this minimum number is at least $1$). It follows that the generating function of any co-Hamiltonian isotopy is a constant function along it orbits. Therefore, we study the co-Hofer norms for co-Hamiltonian isotopies, and establish several co-Hamiltonian and almost co-Hamiltonian ana…
In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction on $M$. We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov's non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylin…
We reconsider the (rational) Calogero-Moser system from the point of view of bi-Hamiltonian geometry. By using geometrical tools of the latter, we explicitly construct set(s) of spectral canonical coordinates, that is, complete sets of Darboux coordinates defined by the eigenvalues and the eigenvectors of the Lax matrix.
We introduce the notion of a hamiltonian 2-form on a Kaehler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kaehler geometry. In particular, on any Kaehler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, and this leads to an explicit description of these Kaehler metrics, which we call weakly Bochner-flat. Hamiltonian 2-forms draw attention to a general construction of Kaehler metrics which interpolates between toric geometry and Calabi-like constructions of metrics on (projective…
This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the following aspects of integrable non-Hamiltonian systems: local and semi-local normal forms and associated torus actions for integrable systems, and the geometry of integrable systems of type $(n,0)$. Most of the results presented in this text are very recent, and some theorems in this text are even original in the sense that they have not been written down explic…
The book can be divided in three parts: the Lagrange geometry of order $k$, presented in the first three chapters, the geometrical theory of the dual manifolds $T^{*k}M$ - chapters 4-7 and the geometry of Hamilton spaces of order $k$ and their subspaces, contained in the last four chapters. They are studied directly and as 'dual' geometry, via Legendre transformation.
We go on with the program started in the companion paper [CDI+] of defining a Poisson bracket structure on the space of solutions of the equations of motion of first order Hamiltonian field theories. The case of non-Abelian gauge theories is addressed by using a suitable version of the coisotropic embedding theorem.
Affine hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational problems for affine hamiltonians and lagrangians of order $k\geq 2$ are studied, relating them to a Hamilton equation. An Ostrogradski type theorem is proved: the Hamilton equation of an affine familtonian $h$ is equivalent with Euler-Lagrange equation of its dual lagrangian $L$. Zermelo condition is also studied and some non-trivial examples are given.
