Ely Kerman
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.
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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. …
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…
Link Floer Homology and a Hofer Pseudometric on Braids
本文受le Roux想法启发，在辫子群上定义Hofer型伪范数，利用Link Floer Homology构造对环绕数敏感的拟态，证明两股辫子情形得到范数并对多股辫子给出下界估计。
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 show that if (M,ω) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer's metric on the group of Hamiltonian diffeomorphisms of (M,ω) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer's metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in M x M when M satisfies the above dynamical condition. To prove this, we use the properties of a Floer-theo…
We use spectral invariants in Lagrangian Floer theory in order to show that there exist \emph{isometric} embeddings of normed linear spaces (finite or infinite dimensional, depending on the case) into the space of Hamiltonian deformations of certain Lagrangian submanifolds in tame symplectically aspherical manifolds. In addition to providing a new class of examples in which the Lagrangian Hofer metric can be computed explicitly, we refine and generalize some known results about it.
Given a closed symplectic manifold $(M,ω)$ we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group ${\hbox{\it Ham}} (M,ω)$ by means of the Hofer metric on ${\hbox{\it Ham}} (M,ω)$. We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold $(M,ω)$ to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we get a new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of products of u…
We prove that the displacement energy of a stable coisotropic submanifold is bounded away from zero if the ambient symplectic manifold is closed, rational and satisfies a mild topological condition.
We construct an embedding $Φ$ of $[0,1]^{\infty}$ 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 $\mathbb{R} \oplus [0,1]^{\infty}$ into either $Ham(M, ω)$ or $\widetilde{Ham}(M, ω)$, 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…
The equivalence of Ekeland-Hofer and equivariant symplectic homology capacities
该研究证明了Ekeland-Hofer容量与等变辛同调容量在$\u211d^{2n}$所有星形域上一致，解决了35年的公开问题，同时建立了等变Floer同调与Morse同调的链复形同构关系。
In this short note we show how Dubrovin's integrable hierarchies, defined using the Gromov-Witten theory of a closed symplectic manifold, generalizes to Hamiltonian Floer theory. In particular, we show how the required generalization of the PSS isomorphism, relating Gromov-Witten theory and Hamiltonian Floer theory, can be constructed in the framework of Eliashberg-Givental-Hofer's symplectic field theory.
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 find robust obstructions to representing a Hamiltonian diffeomorphism as a full $k$-th power, $k \geq 2,$ and in particular, to including it into a one-parameter subgroup. The robustness is understood in the sense of Hofer's metric. Our approach is based on the theory of persistence modules applied in the context of filtered Floer homology. We present applications to geometry and dynamics of Hamiltonian diffeomorphisms.
On the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics
本文研究哈密顿同胚在闭合辛流形上的周期轨道数量问题，证明非收缩周期轨道的霍弗-泽亨德猜想在广泛辛流形类别中成立。
We verify here some variants of topological and dynamical flavor of the injectivity radius conjecture in Hofer geometry, Lalonde-Savelyev \cite{citeLalondeSavelyevOntheinjectivityradiusinHofergeometry} in the case of $Ham (S^2)$ and $Ham(Σ, ω)$, for $Σ$ a closed positive genus surface. In particular we show that any loop in $Ham (S^2)$, respectively $Ham(Σ, ω)$ with $L ^{+}$ Hofer length less than $area(S ^{2} )/2$, respectively any $L ^{+} $ length is contractible through ($L ^{+} $) Hofer shorter loops, in the $C ^{\infty} $ topology. We also prove some stronger variants of this statement on…
In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit a metric of negative sectional curvature. Such Lagrangian submanifolds exist in every symplectic manifold of dimension greater than six or equal to four. The proof utilizes the relations between closed geodesics on the Lagrangian, the periodic orbits of geometric Hamiltonian flow…
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…
Hofer's norm (metric) is an important and interesting topic in symplectic geometry. In the present paper, we define fragmented Hofer's norms which are Hofer's norms controlled by fragmentation norms and give some observations on fragmented Hofer's norms.
On Cap-Decorated Floer Persistence modules
该研究针对闭辛非球面流形，引入带帽装饰的Floer持续模，证明其层的函子性与稳定性，构造通用帽轮廓伪距离，得到哈密顿共轭类新不变量，可检测普通Floer方法无法察觉的辛映射类位移现象。
We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along with a few other geometric situations. We provide sample applications to the $C^0$-geometry of Morse functions and to Hofer's geometry of Hamiltonian diffeomorphisms, that go beyond spectral invariants and traditional persistent homology.
Fredholm notions in scale calculus and Hamiltonian Floer theory
本文在尺度Banach空间上给出Fredholm性质的等价定义，并引入非线性尺度Fredholm性质，该性质更易验证且适用于全纯曲线模空间，并以Hamilton Floer理论中的轨迹断裂为例进行说明。
On the Lagrangian Hofer geometry of Clifford tori
本文证明在复射影空间或四维二次曲面中，与Clifford环面Hamilton同伦的Lagrangian空间在Chekanov Hofer度量下包含实数轴的拟等距副本，从而具有无限直径。
For a class of Riemannian manifolds that include products of arbitrary compact manifolds with manifolds of nonpositive sectional curvature on the one hand, or with certain positive-curvature examples such as spheres of dimension at least 3 and compact semisimple Lie groups on the other, we show that the Hamiltonian diffeomorphism group of the cotangent bundle contains as subgroups infinite-dimensional normed vector spaces that are bi-Lipschitz embedded with respect to Hofer's metric; moreover these subgroups can be taken to consist of diffeomorphisms supported in an arbitrary neighborhood of t…
Stefan M$\ddot{\mathrm{u}}$ller posed the problem "Do Hofer's metrics on the group of Hamiltonian diffeomorphism and the one of Hamiltonian homeomorphisms (Hameomorphisms) correspond?". Let $(M,ω)$ be a compact exact symplectic manifold. We prove that the group of Hamiltonian homeomorphisms is not a simple group if the positive answer of M$\ddot{\mathrm{u}}$ller's question holds . To prove this results, we introduce fragmented Hofer's norms on a normal subgroup of the group of Hamiltonian homeomorphisms.
In this paper we show how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures in Hamiltonian Floer theory. Using the SFT of Hamiltonian mapping tori we show how to define a homotopy extension of the well-known Lie bracket and discuss how it can be used to prove the existence of multiple closed Reeb orbits. Furthermore we show how to define the analogue of rational Gromov-Witten theory in the Hamiltonian Floer theory of open symplectic manifolds. More precisely, we introduce a so-called cohomology F-manifold…
The purpose of this note is to make some connection between the sub-Riemannian geometry on Carnot-Caratheodory groups and symplectic geometry. We shall concentrate here on the Heisenberg group, although it is transparent that almost everything can be done on a general Carnot-Caratheodory group. Such generalisations will be the subject of a forthcoming note.
Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory
本文在特定拓扑条件下比较了闭辛流形及其闭Lagrangian子流形在周期轨道与Lagrangian Floer同论情形下的谱不变量，导出Hofer距离比较推论，并定义了相关乘积证明其不变量的次可加性。
In this note we consider the following conjecture: given any closed symplectic manifold $M$, there is a sufficiently small real positive number $ρ$ such that the open ball of radius $ρ$ in the Hofer metric centered at the identity on the group of Hamiltonian diffeomorphisms of $M$ is contractible, where the retraction takes place in that ball -- this is the strong version of the conjecture -- or inside the ambient group of Hamiltonian diffeomorphisms of $M$ -- this is the weak version of the conjecture. We prove several results that support that weak form of the conjecture.
To every closed subset $X$ of a symplectic manifold $(M,ω)$ we associate a natural group of Hamiltonian diffeomorphisms $Ham(X,ω)$. We equip this group with a semi-norm $\Vert\cdot\Vert^{X,ω}$, generalizing the Hofer norm. We discuss $Ham(X,ω)$ and $\Vert\cdot\Vert^{X,ω}$ if $X$ is a symplectic or isotropic submanifold. The main result involves the relative Hofer diameter of $X$ in $M$. Its first part states that for the unit sphere in $R^{2n}$ this diameter is bounded below by $\frac\pi2$, if $n\geq2$. Its second part states that for $n\geq2$ and $d\geq n+1$ there exists a compact set in $R^{…
Schwarz showed that when a closed symplectic manifold (M,\om) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on π_2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham (M,\om). In this note we describe less stringent conditions on the Chern class and quantum homology of M under which the (asymptotic) spectral invariants descend to Ham (M,\om). For example, they descend if the quantum multiplication of M is undeformed and H_2(M) has rank >1, or if the minimal …
Billiards and Hofer's Geometry
本文建立凸平面台球与辛拓扑中Hofer几何的联系，证明光滑严格凸台球桌的台球球映射间Hofer距离可由台球桌间简单几何距离上界控制，并由此证明凸多边形台球球映射属于圆环光滑保面积映射群在Hofer度量下的完备化。
We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer's metric. Namely: (1) we resolve the Kapovich-Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also obtain, as a corollary, that the group of area-preserving diffeomorphisms of the…
This paper meticulously revisit and study the flux geometry of any compact oriented manifold $(M; W)$. We generalize several well-known factorization results, exhibit some orbital conditions for the study of flux geometry, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, show that any smooth path in the kernel of the flux is a vanishing flux path, and show that the kernel of the flux for volume-preserving diffeomorphisms is $C^1$closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map:This recovers several results fr…
A homological criterion for the almost-existence of Hamiltonian chords
本文提出了基于wrapped Floer同调的判据，通过引入相对coisotropic Hofer-Zehnder容量，证明了当wrapped Floer同调消失时任意超曲面附近Hamiltonian弦的几乎存在性。
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…
Hofer-Like Geometry Revisited
本文证明了Banyaga的霍弗-like范数在紧致辛流形上的哈密顿微分同胚限制下与经典霍弗范数一致，填补了霍弗与霍弗-like几何之间的空白。
We study the class of norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such norm that is continuous with respect to the $C^{\infty}$-topology, is dominated from above by the $L_{\infty}$-norm. As a corollary, we obtain that any bi-invariant Finsler pseudo-metric on the group of Hamiltonian diffeomorphisms that is generated by an invariant norm that satisfies the aforementioned continuity assumption, is either identically zero or equivalent to Hofer's metric.
We prove the following three results in Hamiltonian dynamics. 1. The Weinstein conjecture holds true for every displaceable hypersurface of contact type. 2. Every magnetic flow on a closed Riemannian manifold has contractible closed orbits for a dense set of small energies. 3. Every closed Lagrangian submanifold of an arbitrary symplectic manifold whose fundamental group injects and which admits a Riemannian metric without closed geodesics has the intersection property.
This is the first of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on the symplectic cohomology of open symplectic manifolds. Using the SFT of Hamiltonian mapping tori we show how to define a homotopy extension of the well-known Lie bracket on symplectic cohomology. Apart from discussing applications to the existence of closed Reeb orbits, we outline how the $L_{\infty}$-structure is conjecturally related via mirror symmetry to the extended deformation theory of compl…
In this article we study persistence features of topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces with respect to Hofer's metric. We exhibit stability of these dynamical quantities in a rather strong sense for a specific family of maps introduced by Polterovich and Shelukhin. A crucial ingredient comes from some enhancement of lower bounds for the topological entropy and orbit growth forced by a periodic point, formulated in terms of the geometric self-intersection number and a variant of Turaev's cobracket of the free homotopy class that it induces. Tho…
Hofer geometry of $A_3$-configurations
研究$A_3$配置的霍弗度量中嵌入无限维准平面的性质，揭示哈密顿同胚群中存在无限维准平面。
Complexity of Hofer's geometry in higher dimensional manifolds
本文研究了高维辛流形中哈密顿群的几何复杂性，揭示了哈密顿微分同胚作为k次幂的限制及大尺度几何结构。
In this article we study the Hofer geometry of a compact Lie group $K$ which acts by Hamiltonian diffeomorphisms on a symplectic manifold $M$. Generalized Hofer norms on the Lie algebra of $K$ are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite dimensional Lie groups $K$ endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group $K$ and in the group of Hamiltonian diffeomorphisms. These results a…
Hamiltonian Floer theory on surfaces
本文利用有限能量叶状结构思想，发展了曲面上哈密顿同痕动力学与链级 Floer 理论的联系，提出生成正向横截叶状结构的 Floer 理论方法，并基于 Floer 链的拓扑刻画定义了新型谱不变量。
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.
Symplectic integrators in the realm of Hofer's geometry
研究从哈密顿和李形式构造的辛积分器，利用Hofer几何优化不同辛格式的耦合，得到Parareal方法中哈密顿动力学良好行为的约束。
Lagrangian Floer theory, from geometry to algebra and back again
本文综述 Floer 理论从解决 Arnold 猜想到 Fukaya 范畴代数结构，再到基于镜像对称的族 Floer 上同调几何的发展脉络。
Spectral invariants over the integers
本研究在哈密顿Floer同调框架下探究谱不变量对系数的依赖性，发现$\boldsymbol{\boldsymbol{\boldsymbol{Z}}}$系数下谱范数在复射影空间无界、域系数下一致有界的独有现象，解答了Hingston问题的辛版本，并给出相关动力学与几何应用。
Invariant sets and spectral rigidity of Hamiltonian diffeomorphisms
该研究解决了Polterovich2002年关于二维哈密顿微分同胚Hofer度量刚性的问题，在高维自治哈密顿流等情形取得进展，相关结果适用于Viterbo谱度量等，运用定量拉格朗日Floer理论完成论证。
Hofer geometry via toric degeneration
本文利用环面退化在哈密顿微分同胚群上构造不同的齐次拟态射，研究了二次超曲面和del Pezzo曲面上的拉格朗日子流形，并证明了超重性及相关的辛拓扑应用。
Random Hamiltonians II: A central limit theorem and the Hofer geometry of random walks
本文利用随机游走研究哈密顿微分同胚群的整体几何，证明其随机游走期望霍弗范数增长与步数平方根相关，给出相关测度、可测性及中心极限定理的结果。
We relate previously defined quantum characteristic classes to Morse theoretic aspects of the Hofer length functional on $\ls$. As an application we prove a theorem which can be interpreted as stating that this functional behaves "virtually" as a perfect Morse-Bott functional with a flow. This can be applied to study topology and Hofer geometry of $ \text {Ham}(M, ω)$. We also use this to give a prediction for the index of some geodesics for this functional, which was recently partially verified by Yael Karshon and Jennifer Slimowitz.
In this paper we consider a geometric variant of Hofer's symplectic energy, which was first considered by Eliashberg and Hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary. We prove, by a simple geometric argument, that both versions of energy give rise to genuine norms on all symplectic manifolds. Roughly speaking, we show that if there were a symplectomorphism of $M$ which had "too little" energy, one could embed a large ball into a thin cylinder $M \times B^2$. Thus there is a direct geometric relation betwee…
We study the asymptotic behaviour of 1-parameter subgroups with respect to Hofer's metric when the underlying symplectic manifold is an open surface of infinite area. We prove that, depending on the topology of the level sets of the Hamiltonian H, the distance either is bounded or behaves asymptotically linear. Moreover, the slope can be calculated explicitly as the difference of two distinguished critical values of H.
It is the goal of this paper to present the first steps for defining the analogue of Hamiltonian Floer theory for covariant field theory, treating time and space relativistically. While there already exist a number of competing geometric frameworks for covariant field theory generalizing symplectic geometry, none of them are readily suitable for variational techniques such as Hamiltonian Floer theory, since the corresponding action functionals are too degenerate. Instead, we show how a regularization procedure introduced by Bridges leads to a new geometric framework for which we can show that …
Given a closed monotone symplectic manifold $M$, we define certain characteristic cohomology classes of the free loop space $L \text {Ham}(M, ω)$ with values in $QH_* (M)$, and their $S^1$ equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring $H_{*} (L\text {Ham}(M, ω), \mathbb{Q})$, with its Pontryagin product to $QH_{2n+*} (M)$ with its quantum product. As an application we prove an extension of a theorem of McDuff …
Braid Stability and a Floer theory of Braid Isotopies
本文在闭辛曲面哈密顿流周期轨道辫子空间上建立Floer理论，证明定量辫子稳定性，并应用于二维环面，仅用可缩轨道证明拓扑熵约束集的Hofer距离趋于无穷。
Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings
本文建立Poisson几何与Fukaya范畴之间的联系，以辛群胚解释幺半Fukaya结构及其对辛叶子范畴的作用，并据此说明部分Lagrangian Floer上同调环交换的原因。
From Euclidean field theory to hyperkähler Floer theory via regularized polysymplectic geometry
本研究在Ginzburg和Hein的工作基础上，通过Bridges多辛形式体系的正则化方案，将欧几里得场论与超凯勒Floer理论关联，证明带非线性拉普拉斯方程的周期解问题由超凯勒Floer理论承担，并得到杯长估计结果。
Floer sections in multisymplectic geometry
本文在多辛几何中引入了新的框架，将辛几何中的椭圆方法推广，并证明了伪Fueter曲线的存在。
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