Hamiltonian Dynamics

Spectral numbers in Floer theories, Compositio Mathematica, 144 (2008), no. 6, 1581-1592.

This paper proves a foundational result about an important tool in filtered Floer theory, namely the spectral invariant (developed earlier by Matthias Schwarz, Yong-Geun Oh, and others).  To any class in the homology of the chain complex, one can associate a spectral number defined as the infimal filtration level of any chain representing the homology class.  This paper shows, under purely algebraic hypotheses which model the chain complexes found in many different Floer theories, that this infimum is always attained by some chain in the complex (typically the possible filtration levels form a countable dense subset of the reals, so this is a nontrivial statement).  The usefulness of this result lies in the fact that it sometimes allows one to draw conclusions about the behavior of the spectral numbers as the Hamiltonian is varied in a suitable one-parameter family; in particular the result leads to an important homotopy-invariance property for the spectral numbers.

Floer homology in disc bundles and symplectically twisted geodesic flows, Journal of Modern Dynamics, 3 (2009), no. 1, 61-101.

I prove here that, subject to a Chern class assumption, an autonomous Hamiltonian attaining a Morse-Bott minimum along a closed symplectic submanifold of a symplectic manifold has periodic orbits on all sufficiently low energy levels.  In particular this theorem applies to the motion of a charged particle moving in a magnetic field on a closed manifold, provided that the 2-form representing the field is nondegenerate.  This extends work of Viktor Ginzburg and Basak Gurel, who proved the result when the submanifold is spherically rational; the extension requires novel estimates on Floer trajectories just to set the relevant Floer theory up in the more general context, as well as a detailed understanding of the appropriate Floer complex at the chain level.

Some of the technical ingredients used to set up the Floer theory in this paper later played a role in Doris Hein's extension of the Conley conjecture to manifolds with irrational symplectic forms.

 The sharp energy-capacity inequality, Communications in Contemporary Mathematics, 12 (2010), no. 3, 457-473. (copyright World  Scientific Publishing Company, doi 10.1142/S0219199710003889)

It is shown here that, in any closed or Stein symplectic manifold, the Hofer-Zehnder capacity of a subset (related to the periodic orbits of autonomous Hamiltonians supported in the set) is bounded above by its displacement energy.  In Euclidean space this reduces to an old result of Hofer, but in a more general setting it had only been known up to a constant factor.  The proof exploits a variety of properties of the Oh-Schwarz spectral invariants associated to the fundamental class, including a new result which lets one read off the spectral invariant of an autonomous Hamiltonian with no nontrivial periodic orbits of period less than one.   Along the way one obtains new proofs of a number of old results, such as the nondegeneracy of Oh's spectral metric.

Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds,  Israel Journal of Mathematics, 184 (2011), 1-57.

This paper introduces a new Floer-theoretic invariant of Hamiltonian diffeomorphisms called the boundary depth, defined as the infimal number b such that any chain c in the image of the boundary operator is the boundary of a chain whose filtration level is at most b larger than that of c.  This quantity turns out to satisfy a number of useful properties, which are used here to show that many new classes of coisotropic submanifolds have positive displacement energy, and also to show that members of a certain class of Hamiltonian diffeomorphisms have infinitely many nontrivial periodic orbits.  This paper also contains a lemma which I have come to regard as philosophically important: the filtered chain isomorphism type of the Floer chain complex of a Hamiltonian is an invariant of the associated element of the universal cover of the Hamiltonian diffeomorphism group.

Duality in filtered Floer-Novikov complexes, Journal of Topology and Analysis, 2 (2010), no. 2, 233-258. (copyright World Scientific Publishing Company, doi 10.1142/S1793525310000331)

There is a natural pairing between the Floer complex of a Hamiltonian diffeomorphism and that of the inverse diffeomorphism, which can be thought of as a Floer-theoretic version of Poincare duality.  Passing to appropriate filtered versions of the Floer groups, one obtains an analogue of the Poincare-Lefschetz duality pairing on a manifold with boundary.  This paper proves that, after passing to homology, this Poincare-Lefschetz pairing is nondegenerate.  If one imposes a discreteness hypothesis on the symplectic form, this is an easy consequence of the universal coefficient theorem, but in a more general setting the universal coefficient theorem does not apply because the relevant complexes are not each others' duals.  The main application of this nondegeneracy result is that it implies a duality relationship for the spectral invariants, which is necessary for the construction of Entov-Polterovich-type Calabi quasimorphisms on some symplectic manifolds.

Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms, Geometry & Topology 15 (2011), 1313-1417.

The Hamiltonian Floer complex admits a pair of pants product which, on passing to homology, recovers the quantum product structure on the homology of the underlying symplectic manifold.  This quantum product structure, meanwhile, has been known for many years to admit a family of deformations parametrized by the even-dimensional homology of the manifold.  This paper lifts these deformations to chain level, yielding a family of deformed Hamiltonian Floer chain complexes each equipped with a deformed product structure which gives the corresponding deformed quantum product on passing to homology.  Unlike the situation for analogous constructions in Lagrangian Floer theory or Morse theory, only the ring structure, not the module structure, of the homology depends on the deformation.  These deformations allow one to connect the Entov-Polterovich constructions of Calabi quasimorphisms to the theory of "big quantum homology" as studied by algebraic geometers, yielding Calabi quasimorphisms on all symplectic toric manifolds and on point blowups of all symplectic manifolds.  Also, one obtains a Floer-theoretic proof of a result of Guangcun Lu about the Hofer-Zehnder capacity of symplectically rationally connected manifolds.

Many closed symplectic manifolds have infinite Hofer-Zehnder capacity, to appear in Transactions of the American Mathematical Society.

This paper exhibits a multitude of closed symplectic manifolds which carry autonomous Hamiltonian systems with no nontrivial periodic orbits.  The main observation is that, subject to an often-satisfied topological condition, a manifold formed by symplectic sum along a torus contains a hypersurface (diffeomorphic to a principal circle bundle over the torus) which can be made to have no closed characteristics under a perturbation of the symplectic form.  Since the symplectic sum can be used to construct many interesting symplectic manifolds, especially in dimension four, one  gets many examples this way, including elliptic surfaces and  the symplectic four-manifolds constructed by Gompf having arbitrary fundamental group.  All of the four-dimensional examples have b^+>1; indeed when b^+=1 results from Seiberg-Witten theory imply that such aperiodic symplectic forms cannot exist.

Hofer’s metrics and boundary depth (revised May 2012)

There is a natural biinvariant metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold constructed by Hofer. If one considers the action of the group of Hamiltonian diffeomorphisms on the set of Lagrangian submanifolds, Hofer’s metric induces a nondegenerate metric on each of the orbits of the action, as shown by Chekanov. This paper exploits the properties of the boundary depth to show that, in a variety of cases, these metric spaces are quite large in the sense that they admit quasi-isometric embeddings of infinite-dimensional normed vector spaces. The key technical observation is that the boundary depth gives a function that is well-defined on the Hamiltonian diffeomorphism group (or, in Lagrangian Floer theory, on pairs of Lagrangian submanifolds), not just on the universal cover, so one can get lower bounds on the Hofer metric using Floer theory without needing to study issues of monodromy. It’s still not known whether it always holds that the Hamiltonian diffeomorphism group of a symplectic manifold has infinite Hofer diameter; this paper ends with a simple argument showing that the Chekanov-Hofer metric on Lagrangian submanifolds of R² Hamiltonian-isotopic to the unit circle has finite diameter.

Submanifolds and the Hofer norm

Hofer's metric on the Hamiltonian diffeomorphism group induces a pseudometric on the orbit of any closed submanifold under the group. Chekanov showed that this pseudometric is a genuine metric when the submanifold is a compact Lagrangian, and this paper addresses more general submanifolds. The general conclusion is that the behavior of the pseudometric has much to do with "how coisotropic" the submanifold is. In particular an explicit construction shows that the submanifold must be (everywhere) coisotropic for the pseudometric to be nondegenerate, and conversely it is shown that many classes of coisotropic submanifolds (including all hypersurfaces) do indeed have nondegenerate Chekanov-Hofer metrics. On the other hand, I find that for any codimension larger than one the image of a generic embedding (and in particular of any nowhere coisotropic embedding) has Chekanov-Hofer metric which vanishes identically (i.e. the image is "weightless"). When the image of such an embedding is displaceable it follows that it has zero displacement energy, yielding examples of submanifolds which have zero displacement energy but no nonvanishing normal vector fields. A key step is the identification of a subset of the submanifold called the rigid locus which is found (with the help of Banyaga's fragmentation lemma) to contain much information about the pseudometric. This in particular makes it possible to show that "weightlessness" will follow under a condition that can be studied using the jet transversality theorem.