Hamiltonian Dynamics and Morse theory

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, Transactions of the American Mathematical Society 364 (2012), no. 11, 5913-5943.

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, Annales Scientifiques de l'École Normale Supérieure 46 (2013), no. 1, 57-128.

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, Journal of the European Mathematical Society 16 (2014), no. 8, 1571-1616.

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.

Linking and the Morse complex, Annales de la Faculté des Sciences de Toulouse 23 (2014), no. 1, 25-94.

Various classic theorems of Rabinowitz and his collaborators show that, in quite general situations, one can deduce the existence of a critical point of a function f from the presence of a pair of linked submanifolds such that f takes higher values on one of the submanifolds than on the other. For the special case of a Morse function on a compact manifold, this paper obtains a more refined result, showing that f has more critical points than are required by the Morse inequalities if and only if there is a certain type of link with nontrivial linking number that is separated by f, and indeed giving a precise characterization of the number of critical points of f in terms of linking numbers of links and gradient flow chords connecting their components. The proofs make use of various operations on the Morse complex of f which are motivated in part by similar constructions in Floer theory. Along the way, we obtain a purely geometric formula for the Morse-theoretic boundary depth which is manifestly continuous with respect to C⁰ variations in the Morse function.

Hofer geometry and cotangent fibers, Journal of Symplectic Geometry 12 (2014), no. 3, 619-656.

Complementing earlier work which showed that the Hamiltonian diffeomorphism groups of many closed symplectic manifolds admit infinite-dimensional normed vector spaces which are bi-Lipschitz embedded with respect to the Hofer metric, this paper establishes a similar result for many cotangent bundles. To establish the necessary lower bounds, we use the boundary depth of the Lagrangian Floer complex associated to the conormal bundle of a totally geodesic submanifold Q together with the image of cotangent fiber over a point x under a compactly supported Hamiltonian diffeomorphism. For our family of Hamiltonian diffeomorphisms we use certain reparametrizations of the geodesic flow. Although the homology of the Floer complex vanishes, we show that the boundary depths associated to a wide variety of these diffeomorphisms are large provided that the manifold satisfies a certain condition on its geodesics; roughly speaking one needs there to be no Morse-index-one geodesics from the submanifold Q to the point x. This condition can be arranged to hold for a variety of underlying manifolds, including those with nonpositive curvature as well as many positively-curved symmetric spaces; moreover the condition is preserved under taking a product with an arbitrary manifold.

On certain Lagrangian submanifolds of S² x S² and CPⁿ (with Joel Oakley), Algebraic and Geometric Topology 16 (2016), no. 1, 149-209.

This paper had its origins in an attempt to clarify the relations between various constructions of monotone Lagrangian tori in S²xS² and CP², of which one involves the geodesic flow on S² (following Polterovich and Albers-Frauenfelder), another involves toric degeneration models (following Fukaya-Oh-Ohta-Ono and Wu), and another involves Chekanov's twist tori. We show that, in either S²xS² and CP², the tori obtained in these ways are all Hamiltonian isotopic to each other; for S²xS² this had been widely expected due in part to an alternative construction of Entov and Polterovich, while in CP² it had been an open question. We extend our methods to CPⁿ for larger n (and also to quadrics in CPⁿ, generalizing S²xS²) giving monotone Lagrangians in these manifolds that had not previously been considered. Surprsingly, many of these monotone Lagrangian submanifolds are displaceable, unlike previous examples of monotone Lagrangian submanifolds of simply connected closed symplectic manifolds. One way of constructing our submanifolds is by the Biran circle bundle construction, and we give a general topological criterion for a monotone Lagrangian submanifold obtained by this construction to be displaceable, though it remains unclear to what extent this criterion applies to examples beyond the rather specific ones considered in this paper.

Observations on the Hofer distance between closed subsets , Mathematical Research Letters 22 (2015), no. 6, 1805-1820.

This paper proves some elementary but surprising properties of the Hofer distance between two subsets. The first half of the paper was motivated by energy-capacity-type inequalities appearing in recent work of Borman-McLean and Humiliere-Leclercq-Seyfaddini, which showed that for certain Lagrangian submanifolds L, in order for the time-one flow of a Hamiltonian function H to disjoin L from an open subset that it intersects, the time-integral of the oscillation of H when restricted to L must exceed a certain lower bound. The fact that there would be a lower bound for the integral of the oscillation of H over the whole manifold (i.e. a lower bound for the Hofer norm) is rather standard, but the fact that it would suffice to restrict to L seems counterintuitive. But here I explain (and generalize) this improvement by means of a short argument which shows that the (Chekanov-)Hofer distance between any two closed subsets A and B is equal to the infimal time-integral of the oscillation of H restricted to A, over all Hamiltonians H with time-one flow mapping A to B. The second half of the paper proves some new results about the rigid locus of a closed subset of a symplectic manifold, as defined in my earlier JEMS paper. I show that the rigid locus (though not necessarily a submanifold) is always coisotropic in the generalized sense that the ideal of functions vanishing on it is closed under the Poisson bracket, and moreover cannot be properly contained in a half-dimensional submanifold. These results make it possible to show that the Chekanov-Hofer pseudometric vanishes identically on the orbit of any non-Lagrangian, half-dimensional submanifold, or any (possibly singular) complex analytic subvariety of a Kahler manifold.

Persistent homology and Floer-Novikov theory (with Jun Zhang), Geometry & Topology 20 (2016), no. 6, 3333-3430.

We construct "barcodes" for the chain complexes over Novikov rings that arise in Novikov's Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology. Our barcodes completely characterize the filtered chain homotopy type of the chain complex; in particular they subsume in a natural way previous filtered Floer-theoretic invariants such as boundary depth and torsion exponents, and also reflect information about spectral invariants. We moreover prove a continuity result which is a natural analogue both of the classical bottleneck stability theorem in persistent homology and of standard continuity results for spectral invariants, and we use this to prove a C⁰-robustness result for the fixed points of Hamiltonian diffeomorphisms. Our approach, which is rather different from the standard methods of persistent homology, is based on a non-Archimedean singular value decomposition for the boundary operator of the chain complex.

Graphicality, C⁰-convergence, and the Calabi homomorphism, to appear in Bulletin of the Korean Mathematical Society.

Consider a sequence of compactly supported Hamiltonian diffeomorphisms f_k of an exact symplectic manifold, all of which are "graphical" in the sense that their graphs are identified by a Darboux-Weinstein chart with the image of a one-form. We show by an elementary argument that if the f_k C⁰-converge to the identity then their Calabi invariants converge to zero. This generalizes a result of Oh, in which the ambient manifold was the two-disk and an additional assumption was made on the Hamiltonians generating the f_k. We discuss connections to the open problem of whether the Calabi homomorphism extends to the Hamiltonian homeomorphism group. The proof is based on a relationship between the Calabi invariant of a C⁰-small Hamiltonian diffeomorphism and the generalized phase function of its graph.

Symplectically knotted codimension-zero embeddings of domains in R⁴ (with Jean Gutt).

This paper shows that many domains X in R⁴ admit symplectic embeddings into dilates tX of themselves that are "knotted" in the sense that there is no symplectomorphism of the codomain that maps the image of the embedding back to the domain. This contrasts with a result of McDuff that shows that symplectic embeddings of four-dimensional ellipsoids into other ellipsoids can never be knotted. Our examples include a wide variety of convex and concave toric domains, including many polydisks and also some domains that are arbitrarily close to ellipsoids. Typically the examples are produced by using machinery derived by McDuff and others from Taubes-Seiberg-Witten theory to obtain an embedding from an ellipsoid to the desired codomain tX, and then restricting this embedding to the domain X. We use filtered S¹-equivariant symplectic homology to distinguish this restriction from the inclusion of X into tX, the point being that (if t is small enough, depending on X) the filtered S¹-equivariant symplectic homology of an ellipsoid is too simple for the map induced by an unknotted embedding to factor through it.