arXiv research

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-cons…

Defines Lorentzian distance on contactomorphisms, proving continuity and finite conditions.

We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, ω, τ)$ with nonempty and compact real part $L={\rm Fix}(τ)$. For given $Λ\in (0, +\infty]$ and $m\in\N\cup\{0\}$ we show the equivalence of the following two claims: (i) $\sharp(L\capφ^H_1(L))\ge m$ for any Hamiltonia…

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 res…

Study of submanifolds in symplectic and contact manifolds using Hausdorff metrics.

Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.

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 giv…

We define the symplectic displacement energy of a non-empty subset of a compact symplectic manifold as the infimum of the Hofer-like norm [5] of symplectic diffeomorphisms that displace the set. We show that this energy (like the usual displacement energy defined using Hamiltonian diffeomorphisms) is a strictly positiv…

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 restrictio…

Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.

Book introduces Hofer's metric on symplectic diffeomorphisms.

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…

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 …

We prove that the group of Hamiltonian diffeomorphisms of the 2-sphere has infinite diameter with respect to Hofer's metric. Our approach is based on the theory of Lagrangian intersections.

Researchers calculate Hofer-Zehnder capacity for twisted tangent bundles over surfaces.

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 su…

We show, by an elementary and explicit construction, that the group of Hamiltonian diffeomorphisms of certain symplectic manifolds, endowed with Hofer's metric, contains subgroups quasi-isometric to Euclidean spaces of arbitrary dimension.

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 lo…

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.

Paper shows pseudometrics on braid groups are nondegenerate.

Cosymplectic geometry can be viewed as an odd dimensional counterpart of symplectic geometry. Likely in the symplectic case, a related property which is preservation of closed forms $ω$ and $η$, refers to the theoretical possibility of further understanding a cosymplectic manifold $(M, ω, η)$ from its group of diffeomo…

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.

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 n…

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 orde…

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 $H$, consid…

An analogue of the Hofer metric $\varrho_H$ on the Hamiltonian group $Ham(M,Λ)$ of a Poisson manifold $(M,Λ)$ can be defined but there is the problem of its non-degeneracy. First we observe that $\varrho_H$ is a genuine metric on $Ham(M,Λ)$ when the union of all closed leaves (as subsets of $M$) of the corresponding sy…

Study shows convergence of Lagrangian submanifolds under certain metrics.

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 c…

Study on volume continuity of Lagrangian submanifolds.

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 capacitie…

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [Y Eliashberg, A Givental and H Hofer, Introduction to Symplectic Field Theory, Geom. Funct. Anal. Special Volume, Part II (2000) 560--673]. We prove compactness results for moduli spaces of holomorphic curves arising in…

Study spectral invariants over integers, discovering unboundedness and field-dependence.

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 t…

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 s…

In this paper we study submanifolds of contact manifolds. The main submanifolds we are interested in are contact coisotropic submanifolds. Based on a correspondence between symplectic and contact coisotropic submanifolds, we can show contact coisotropic submanifolds admit a $C^0$-rigidity, similar to Humilière-Leclercq…

Study relates symplectic homology capacity to periodic orbits in Liouville domains.

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 (thi…

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 …

This paper introduces a new distance metric for filtered A-infinity categories, focusing on Lagrangian submanifolds.

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.

Given a manifold $M$ endowed with a contact 1-form $α$, a bi-invariant pseudo-metric $\varrho_α$ is introduced on $Cont(M,α)$, the compactly supported identity component of the group of all strict contactomorphisms of $(M,α)$. For $M$ open $\varrho_α$ is a metric.

In this note we observe that while all overtwisted contact structures on compact 3--manifolds are supported by planar open book decompositions, not all contact structures are. This has relevance to invariants of contact structures and also to the Weinstein conjecture via work of Abbas Cieliebak and Hofer.

We give an algebro-geometric construction of some of the non-arithmetic ball quotients constructed by the author, Parker and Paupert. The new construction reveals a relationship between the corresponding orbifold fundamental groups and the automorphism group of the Klein quartic, and also with groups constructed by Bar…

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 billiar…

Paper studies Lagrangian submanifolds and their homological monodromy.

We use Bott periodicity to relate previously defined quantum classes to certain "exotic Chern classes" on $BU$. This provides an interesting computational and theoretical framework for some Gromov-Witten invariants connected with cohomological field theories. This framework has applications to study of higher dimension…

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, ca…

On every compact, orientable, irreducible 3-manifold V which is toroidal or has torus boundary components we construct a contact 1-form whose Reeb vector field R does not have any contractible periodic orbits and is tangent to the boundary. Moreover, if bdry V is nonempty, then the Reeb vector field R is transverse to …