This note discusses some geometrically defined seminorms on the group $\Ham(M, ω)$ of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M, ω)$, giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm. As a consequence we show that if an element in $\Ham(M, ω)$ is sufficiently close to the identity in the $C^{2}$-topology then it may be joined to the identity by a path whose Hofer length is minimal among all paths, not just among paths in the same homotopy class relative to endpoints. Thus, true geodesics always exist for the Hofer norm. The main step in the proof is to show that a "weighted" version of the nonsqueezing theorem holds for all fibrations over $S^2$ generated by sufficiently short loops.Further, an example is given showing that the Hofer norm may differ from the sum of the one sided seminorms.
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.
We remark that, as in the symplectic case, the Hofer norm on the Hamiltonian group of a Poisson manifold is non-degenerate. The proof is a straightforward application of tools from symplectic topology.
We show that commutators of Hamiltonian diffeomorphisms may have arbitrarily large Hofer norm. The proposed technique is applicable to positive genus surfaces and their products. This gives partial answer to a question by McDuff and Polterovich.
There is a number of known constructions of quasimorphisms on Hamiltonian groups. We show that on surfaces many of these quasimorphisms are not compatible with the Hofer norm in a sense they are not continuous and not Lipschitz. The only exception known to the author is the Calabi quasimorphism on a sphere and the induced quasimorphisms on genus-zero surfaces.
Chekanov showed that the Hofer norm on the Hamiltonian diffeomorphism group of a geometrically bounded symplectic manifold induces a nondegenerate metric on the orbit of any compact Lagrangian submanifold under the group. In this paper we consider the orbits of more general submanifolds. We show that, for the Chekanov-Hofer pseudometric on the orbit of a closed submanifold to be a genuine metric, it is necessary for the submanifold to be coisotropic, and we show that this condition is sufficient under various additional geometric assumptions. At the other extreme, we show that the image of a g…
Using a "Hodge decomposition" of symplectic isotopies on a compact symplectic manifold $(M,ω)$, we construct a norm on the identity component in the group of all symplectic diffeomorphisms of $(M,ω)$ whose restriction to the group $Ham(M,ω)$ of hamiltonian diffeomorphisms is bounded from above by the Hofer norm. Moreover, $Ham(M,ω)$ is closed in $Symp(M,ω)$ equipped with the topology induced by the extended norm. We give an application to the $C^0$ symplectic topology. We also discuss extensions of Oh's spectral distance.
We show that the $L^{\infty}$-norm of the contact Hamiltonian induces a non-degenerate right-invariant metric on the group of contactomorphisms of any closed contact manifold. This contact Hofer metric is not left-invariant, but rather depends naturally on the choice of a contact form $α,$ whence its restriction to the subgroup of $α$-strict contactomorphisms is bi-invariant. The non-degeneracy of this metric follows from an analogue of the energy-capacity inequality. We show furthermore that this metric has infinite diameter in a number of cases by investigating its relations to previously de…
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^{…
Let D be a non-displaceable disk in an annulus A. Suppose that g is a compactly supported Hamiltonian which preserves D with translation number n. We show that Hofer's norm |g| is bounded from below by cn for a certain constant c. We also give example of such Hamiltonian which is more efficient than the obvious rotation of D.
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 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…
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.
The group Hameo (M,ω) of Hamiltonian homeomorphisms of a connected symplectic manifold (M,ω) was defined and studied in [7] and further in [6]. In these papers, the authors consistently used the L^{(1,\infty)}-Hofer norm (and not the L^\infty-Hofer norm) on the space of Hamiltonian paths (see below for the definitions). A justification for this choice was given in [7]. In this article we study the L^\infty-case. In view of the fact that the Hofer norm on the group Ham (M,ω) of Hamiltonian diffeomorphisms does not depend on the choice of the L^{(1,\infty)}-norm vs. the L^\infty-norm [9], Y.-G. …
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 es…
Let M be a closed symplectic manifold, and let | | be a norm on the space of all smooth functions on M, which are zero-mean normalized with respect to the canonical volume form. We show that if | | is dominated from above by the L-Infinity-norm, and | | is invariant under the action of Hamiltonian diffeomorphisms, then it is also invariant under all volume preserving diffeomorphisms. We also prove that if | | is, additionally, not equivalent to the L-Infinity-norm, then the induced Finsler metric on the group of Hamiltonian diffeomorphisms on M vanishes identically.
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 Hamiltonian function $H\in C_0^\infty([0, 1]\times M)$ with Hofer's norm $\|H\|<Λ$; (ii) $\sharp {\cal P}(H,τ)\ge m$ for every $H\in C^\infty_0(\R/\Z\times M)$ satisfying $H(t,x)=H(-t,τ(x))\;\forall (t,x)\in\mathbb{R}\times M$ and with Hofer's norm $\|H\|<2Λ$, where ${\cal P}(H, τ)$ is the…
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…
In this article we study the behavior of the Oh-Schwarz spectral invariants under C^0-small perturbations of the Hamiltonian flow. We obtain an estimate, which under certain assumptions, relates the spectral invariants of a Hamiltonian to the C^0-distance of its flow from the identity. Using the mentioned estimate we show that, unlike the Hofer norm, the spectral norm is C^0-continuous on surfaces. We also present applications of the above results to the theory of Calabi quasimorphisms and improve a result of Entov, Polterovich and Py. In the final section of the paper we use our results to an…
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.…
