We use the criteria of Lalonde and McDuff to show that a path that is generated by a generic autonomous Hamiltonian is length minimizing with respect to the Hofer norm among all homotopic paths provided that it induces no non-constant closed trajectories in M. This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general M uses Liu-Tian's construction of S^1-invariant virtual moduli cycles. As a corollary, we find that any semifree action of S^1 on M gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of M. We also establish a version of the area-capacity inequality for quasicylinders.
Motivated by Pazit Haim-Kislev's combinatorial formula for the Ekeland-Hofer-Zehnder capacities of convex polytopes, we give corresponding formulas for $Ψ$-Ekeland-Hofer-Zehnder and coisotropic Ekeland-Hofer-Zehnder capacities of convex polytopes introduced by the second named author and others recently. Contrary to Pazit Haim-Kislev's subadditivity result for the Ekeland-Hofer-Zehnder capacities of convex domains, we show that the coisotropic Hofer-Zehnder capacities satisfy the subadditivity for suitable hyperplane cuts of two-dimensional convex domains in the reverse direction.
We prove representation formulas for the coisotropic Hofer-Zehnder capacities of bounded convex domains with special coisotropic submanifolds and the leaf relation (introduced by Lisi and Rieser recently), study their estimates and relations with the Hofer-Zehnder capacity,give some interesting corollaries, and also obtain corresponding versions of a Brunn-Minkowski type inequality by Artstein-Avidan and Ostrover and a theorem by Evgeni Neduv.
We find bounds for the Hofer-Zehnder capacity of coadjoint orbits of compact Lie groups with respect to the Kostant--Kirillov--Souriau symplectic form in terms of the combinatorics of their Bruhat graph. We show that our bounds are sharp for coadjoint orbits of the unitary group and equal to the diameter of a weighted Cayley graph.
We prove the finiteness of the Hofer-Zehnder capacity of unit disk cotangent bundles of closed Riemannian manifolds, under some simple topological assumptions on the manifolds. The key ingredient of the proof is a computation of the pair-of-pants product on Floer homology of cotangent bundles. We reduce it to a simple computation of the loop product, making use of results of A.Abbondandolo- M.Schwarz.
Let $(M,ω)$ be an aspherical symplectic manifold, which is closed or convex. Let $U$ be an open set in $M$, which admits a circle action generated by an autonomous Hamiltonian $H \in C^\infty(U)$, such that each orbit of the circle action is not contractible in $M$. Under these assumptions, we prove that the Hofer-Zehnder capacity of $U$ is bounded by the Hofer norm of $H$. The proof uses a variant of the energy-capacity inequality, which is proved by the theory of action selectors.
We introduce the notion of a symplectic capacity relative to a coisotropic submanifold of a symplectic manifold, and we construct two examples of such capacities through modifications of the Hofer-Zehnder capacity. As a consequence, we obtain a non-squeezing theorem for symplectic embeddings relative to coisotropic constraints and existence results for leafwise chords on energy surfaces.
We introduce the concept of Hofer-Zehnder $G$-semicapacity (or $G$-sensitive Hofer-Zehnder capacity) and prove that given a geometrically bounded symplectic manifold $(M,ω)$ and an open subset $N \subset M$ endowed with a Hamiltonian free circle action $ϕ$ then $N$ has bounded Hofer-Zehnder $G_ϕ$-semicapacity, where $G_ϕ\subset π_1(N)$ is the subgroup generated by the homotopy class of the orbits of $ϕ$. In particular, $N$ has bounded Hofer-Zehnder capacity. We give two types of applications of the main result. Firstly, we prove that the cotangent bundle of a compact manifold endowed with a fr…
Let $M$ be a compact manifold with an effective semi-free circle action whose fixed point set has trivial normal bundle. We prove that its cotangent bundle endowed with the canonical symplectic form has bounded Hofer-Zehnder sensitive capacity. We give several examples like the product of any compact manifold with $§^n$ or a connected sum $\CP^n # ... # \CP^n$.
We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite ($π_1$-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds near closed symplectic submanifolds in any symplectic manifold.
The Hofer-Zehnder theorem states that almost every hypersurface in a thickening of a hypersurface $S$ in a symplectic manifold $(M,ω)$ carries a closed characteristic provided that $S$ bounds a compact submanifold and $(M,ω)$ has finite capacity. We show that it is enough to assume that the thickening of $S$ has finite capacity.
We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the homology of the manifold, then it must have an infinite number of contractible periodic points. This constitutes a higher-dimensional homological generalization of a celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in 1994.
We prove that if M is a closed, connected, oriented, rationally inessential manifold, then the Hofer-Zehnder capacity of the unit disk bundle of the cotangent bundle of M is finite.
We give a different and simpler proof of a slightly modified (and weaker) variant of a recent theorem of Shelukhin extending Franks' "two-or-infinitely-many" theorem to Hamiltonian diffeomorphisms in higher dimensions and establishing a sufficiently general case of the Hofer-Zehnder conjecture. A few ingredients of our proof are common with Shelukhin's original argument, the key of which is Seidel's equivariant pair-of-pants product, but the new proof highlights a different aspect of the periodic orbit dynamics of Hamiltonian diffeomorphisms.
We use the criteria of Lalonde and McDuff to determine a new class of examples of length minimizing paths in the group $Ham(M)$. For a compact symplectic manifold $M$ of dimension two or four, we show that a path in $Ham(M)$, generated by an autonomous Hamiltonian and starting at the identity, which induces no non-constant closed trajectories of points in $M$, is length minimizing among homotopic paths. The major step in the proof involves determining an upper bound for the Hofer-Zehnder capacity for symplectic manifolds of the type $(M \times D(a))$ where $M$ is compact and has dimension two …
We use Floer homology to study the Hofer-Zehnder capacity of neighborhoods near a closed symplectic submanifold M of a geometrically bounded and symplectically aspherical ambient manifold. We prove that, when the unit normal bundle of M is homologically trivial in degree dim(M) (for example, if codim(M) > dim(M)), a refined version of the Hofer-Zehnder capacity is finite for all open sets close enough to M. We compute this capacity for certain tubular neighborhoods of M by using a squeezing argument in which the algebraic framework of Floer theory is used to detect nontrivial periodic orbits. …
We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in $\mathbb{C}\text{P}^d$ of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer-Zehnder conjecture in the non-degenerated case: every Hamiltonian diffeomorphism of $\mathbb{C}\text{P}^d$ that has at least $d+2$ non-degenerated periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of $J$-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type ineq…
In this article we examine under which conditions symplectic homology with local coefficients of a unit disk bundle $D^*M$ vanishes. For instance this is the case if the Hurewicz map $π_2(M)\to H_2(M;\mathbb{Z})$ is nonzero. As an application we prove finiteness of the $π_1$-sensitive Hofer-Zehnder capacity of unit disk bundles in these cases. We also prove uniruledness for such cotangent bundles. Moreover, we find an obstruction to the existence of $H$-space structures on general topological spaces, formulated in terms of local systems.
We introduce the Hofer-Zehnder $G$-semicapacity $c_{HZ}^G(M,\om)$ of a symplectic manifold $(M,\om)$ with respect to a subgroup $G \subset π_1(M)$ ($c_{HZ}(M,\om) \leq c^G_{HZ}(M,\om)$) and prove that if $(M,\om)$ is tame and there exists an open subset $U \subset M$ admitting a Hamiltonian free circle action with order greater than two then $U$ has bounded Hofer-Zehnder $G$-semicapacity, where $G \subset π_1(M)$ is the subgroup generated by the orbits of the action, provided that the index of rationality of $(M,\om)$ is sufficiently great (for instance, if $[\om]|_{π_2(M)}=0$). We give a lot …
We study the relationship between a homological capacity $c_{\mathrm{SH}^+}(W)$ for Liouville domains $W$ defined using positive symplectic homology and the existence of periodic orbits for Hamiltonian systems on $W$: If the positive symplectic homology of $W$ is non-zero, then the capacity yields a finite upper bound to the $π_1$-sensitive Hofer-Zehnder capacity of $W$ relative to its skeleton and a certain class of Hamiltonian diffeomorphisms of $W$ has infinitely many non-trivial contractible periodic points. En passant, we give an upper bound for the spectral capacity of $W$ in terms of th…
The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We give examples of functions…
We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2) with an irrational symplectic structure). The underlying smooth manifolds of our examples include, for instance: the K3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it; infinitely many minimal four-manifolds having any given finitely-presented group as their fundamental group; and simply connected minimal four-manif…
Let $γ$ be a non-degenerate Ustilovsky geodesic in $Ham (M, ω)$ generated by $H$. We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of $ γ$, as a critical point of the Hofer length functional, with the Conley Zehnder index of the extremizers of $H$, considered as periodic orbits.
We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions, then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic u…
Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also compute the Gromov width and the Hofer--Zehnder capacity for Cartan domains and their products.
We study geodesics of Hofer's metric on the space of Lagrangian submanifolds in arbitrary symplectic manifolds from the variational point of view. We give a characterization of length-critical paths with respect to this metric. As a result, we prove that if two Lagrangian submanifolds are disjoint then we cannot join them by length-minimizing geodesics.
This paper defines symplectic scale manifolds based on Hofer-Wysocki-Zehnder's scale calculus. We introduce Hamiltonian vector fields and flows on these by narrowing down sc-smoothness to what we denote by strong sc-smoothness, a concept which effectively formalizes the desired smoothness properties for Hamiltonian functions. We show the concept to be invariant under sc-smooth symplectomorphisms, whence it is compatible with Hofer's scale manifolds. We develop and verify the theory at the hand of the free Schrödinger equation.
Scale structures were introduced by H.Hofer, K.Wysocki, and E.Zehnder. In this note we define an invariant for scale Hilbert spaces modulo scale isomorphism and use it to distinguish large classes of scale Hilbert spaces.
Scale spaces were defined by H.Hofer, K.Wysocki, and E.Zehnder. In this note we introduce a subclass of scale spaces and explain why we believe that this subclass is the right class for a general setup of Floer theory.
This paper continues to carry out a foundational study of Banyaga topologies of a closed symplectic manifold [3]. Our intension in writing this paper is to provide several symplectic analogues of some results found in the study of Hamiltonian dynamics. Especially, without appealing to the positivity of the symplectic displacement energy, we point out the impact of the $L^\infty$ version of Banyaga Hofer-like metric in the investigation of the symplectic nature of the $C^0-$limit of a sequence of symplectic maps. This result is the symplectic analogue of a result that was proved in Hofer-Zehnde…
For many classes of symplectic manifolds, the Hamiltonian flow of a function with sufficiently large variation must have a fast periodic orbit. This principle is the base of the notion of Hofer-Zehnder capacity and some other symplectic invariants and leads to numerous results concerning existence of periodic orbits of Hamiltonian flows. Along these lines, we show that given a negatively curved manifold M, a neigbhourhood U of M in the cotangent bundle, a sufficiently small magnetic field and a non-trivial free homotopy class of loops, then the magnetic flow of a Hamiltonian with big enough va…
In this paper we first apply the chain level Floer theory to the study of Hofer's geometry of Hamiltonian diffeomorphism group in the cases without quantum contribution: we prove that any quasi-autonomous Hamiltonian path on weakly exact symplectic manifolds or any autonomous Hamiltonian path on arbitrary symplectic manifolds is length minimizing in its homotopy class with fixed ends, as long as it has a fixed maximum and a fixed minimum which are not over-twisted and all of its contractible periodic orbits of period less than one are sufficiently $C^1$-small. Next we give a construction of ne…
On any closed symplectic manifold we construct a path-connected neighborhood of the identity in the Hamiltonian diffeomorphism group with the property that each Hamiltonian diffeomorphism in this neighborhood admits a Hofer and spectral length minimizing path to the identity. This neighborhood is open in the $C^1-$topology. The construction utilizes a continuation argument and chain level result in the Floer theory of Lagrangian intersections.
In this paper, we use Floer theory to study the Hofer length functional for paths of Hamiltonian diffeomorphisms which are sufficiently short. In particular, the length minimizing properties of a short Hamiltonian path are related to the properties and number of its periodic orbits.
Scale structures were introduced by H. Hofer, K. Wysocki, and E. Zehnder as a new concept of a smooth structure in infinite dimensions. We prove that scale structures on mapping spaces are completely determined by the dimension of domain manifolds. As a consequence, we give a complete description of the local invariant introduced by U. Frauenfelder for mapping spaces. Product mapping spaces and relative mapping spaces are also studied. Our approach is based on the spectral resolution of Laplace type operators together with the eigenvalue growth estimate.
In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiards.
We improve the estimates for the Ekeland--Hofer--Zehnder capacity of convex bodies by Gluskin and Ostrover. In the course of our argument we show that a closed characteristic of minimal action on the boundary of a centrally symmetric convex body in $\mathbb R^{2n}$ must itself be centrally symmetric, with generalizations to some other types of symmetry.
The main purpose of this paper is to study the length minimizing property of Hamiltonian paths on closed symplectic manifolds $(M,ω)$ such that there are no spherical homology class $A \in H_2(M)$ with $$ ω(A) > 0 \quad \text{and} \quad -n \leq c_1(A) < 0, $$ which we call {\it very strongly semi-positive}. We introduce the notion of {\it positively $μ$-undertwisted} Hamiltonian paths and prove that any positively undertwisted quasi-autonomous Hamiltonian path is length minimizing in its homotopy class as long as it has a fixed maximum and a fixed minimum point that are generically under-twist…
In this paper we provide a criterion for the quasi-autonomous Hamiltonian path (``Hofer's geodesic'') on arbitrary closed symplectic manifolds $(M,ω)$ to be length minimizing in its homotopy class in terms of the spectral invariants $ρ(G;1)$ that the author has recently constructed (math.SG/0206092). As an application, we prove that any autonomous Hamiltonian path on arbitrary closed symplectic manifolds is length minimizing in {\it its homotopy class} with fixed ends, when it has no contractible periodic orbits {\it of period one}, has a maximum and a minimum point which are generically under…
In this article we prove that for a smooth fiberwise convex Hamiltonian, the asymptotic Hofer distance from the identity gives a strict upper bound to the value at 0 of Mather's $β$ function, thus providing a negative answer to a question asked by K. Siburg in \cite{Siburg1998}. However, we show that equality holds if one considers the asymptotic distance defined in \cite{Viterbo1992}.
An action selector associates, in a suitable way, to each compactly supported Hamiltonian on a symplectic manifold an action value of the Hamiltonian. Action selectors are known to exist for a broad class of symplectic manifolds. We show how the existence of an action selector leads to sharp energy capacity inequalities between the Gromov width, the Hofer--Zehnder capacity, and the displacement energy. We also obtain sharp lower bounds for the smallest action of a closed characteristic on contact type hypersurfaces.
We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold (M,ω) which upon passing to homology yields ring isomorphisms with the big quantum homology of M. By studying the properties of the resulting deformed version of the Oh-Schwarz spectral invariants, we obtain a Floer-theoretic interpretation of a result of Lu which bounds the Hofer-Zehnder capacity of M when M has a nonzero Gromov-Witten invariant with two point constraints, and we produce a new algebraic criterion for (M,ω) to admit a Calabi quasimo…
