本文将 Shelukhin 证明的 Hofer-Zehnder 猜想同调版本推广至作为辛轨形的加权射影空间，证明了非退化 Hamilton 微分同胚的不动点数超过最小值时必有无穷多周期点。
摘要：
我们将 Shelukhin 证明的 Hofer-Zehnder 猜想同调版本推广到了作为辛轨形的加权射影空间。特别地，我们证明：若此类空间上非退化 Hamilton 微分同胚的以迷向阶为重数计算的不动点数大于可能的最小数目，则存在无穷多个周期点。
英文摘要：
We prove an extension of the homology version of the Hofer-Zehnder conjecture proved by Shelukhin to the weighted projective spaces which are symplectic orbifolds. In particular, we prove that if the number of fixed points counted with their isotropy order as multiplicity of a non-degenerate Hamiltonian diffeomorphism of such a space is larger than the minimum number possible, then there are infinitely many periodic points.
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…
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 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.
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 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…
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.
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 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 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 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…
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…
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.
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.
For any nonempty, compact and fiberwise convex set $K$ in $T^*\mathbb{R}^n$, we prove an isomorphism between symplectic homology of $K$ and a certain relative homology of loop spaces of $\mathbb{R}^n$. We also prove a formula which computes symplectic homology capacity (which is a symplectic capacity defined from symplectic homology) of $K$ using homology of loop spaces. As applications, we prove (i) symplectic homology capacity of any convex body is equal to its Ekeland-Hofer-Zehnder capacity, (ii) a certain subadditivity property of the Hofer-Zehnder capacity, which is a generalization of a …
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.
Helmut Hofer introduced in '93 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3--manifolds $(M,ξ)$ with $π_2(M) \ne 0$. We modify Hofer's argument to prove the Weinstein conjecture for some examples of higher dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit.
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.
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.
We re-prove Gromov's non-squeezing theorem by applying Polyfold Theory to a simple Gromov-Witten moduli space. Thus we demonstrate how to utilize the work of Hofer-Wysocki-Zehnder to give proofs involving moduli spaces of pseudoholomorphic curves that are relatively short and broadly accessible, while also fully detailed and rigorous. We moreover review the polyfold description of Gromov-Witten moduli spaces in the relevant case of spheres with minimal energy and one marked point.
We study here some aspects of the topology of the space of smooth, stable, genus 0 curves in a Riemannian manifold $X$, i.e. the Kontsevich stable curves, which are not necessarily holomorphic. We use the Hofer-Wysocki-Zehnder polyfold structure on this space and some natural characteristic classes, to show that for $X=BU,$ the rational homology of the spherical mapping space injects into the rational homology of the space of stable curves. We also give here a definition of what we call $q$-complete symplectic manifolds, which roughly speaking means Gromov-Witten theory captures all informatio…
Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analytic framework for a variety of geometric elliptic PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to systematically address the common difficulties of compactification and transversality with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. We shine meta-mathematical light on the bigger picture and core ideas of this theory. In addition, we compiled and condensed the core definitions and th…
