arXiv subjects
Publications and source records attributed to Dan Cristofaro-Gardiner.
We show that a simply connected symplectic manifold admitting a complete nonpositively curved compatible metric is symplectomorphic to R^{2n} with its standard symplectic form. This answers the "symplectic Hadamard question" of McDuff-Salamon in the affirmative.
We study the related notions of curvature and perimeter for toric boundaries and their implications for symplectic packing problems in dimension 4; a natural setting for this is a generalized version of convex toric domain which we also study, where there are no conditions on the moment polytope at all aside from convexity. We show that the subleading asymptotics of the ECH and elementary ECH capacities recover the perimeter of such domains in their liminf, without any genericity required, and hence the perimeter is an obstruction to a full filling. As an application, we give the first examples of the failure of packing stability by open subsets of compact manifolds with smooth boundary or with no boundary at all; this has implications for long-term super-recurrence. We also show that a single smooth point of positive curvature on the toric boundary obstructs the existence of an infinite staircase, and we build on this to completely classify smooth (generalized) convex toric domains which have an infinite staircase. We also extend a number of theorems to generalized convex toric domains, in particular the "concave to convex", embedding theorem and the "accumulation point theorem". A curvy point forces "infinite complexity"; we raise the question of whether an infinitely complex domain can ever have an infinite staircase and we give examples with infinite staircases and arbitrarily high finite complexity.
We show that a topological symplectic manifold has a canonically associated bi-Lipschitz structure. As a corollary, we obtain the first examples of non-existence and non-uniqueness for topological symplectic structures. Our arguments hold for any topological manifold admitting an atlas with transition maps that are $C^0$--limits of bi-Lipschitz homeomorphisms.
We survey a number of Weyl type laws that have recently been established in low-dimensional symplectic geometry. These have had a number of applications, which we also introduce. We sketch a number of proofs so that the reader can get a sense of how these formulas are proved and how they can be applied.
These are notes to accompany my lectures at the $2024$ "Current Developments in Mathematics" conference hosted by Harvard/MIT. The lectures were about some recent progress in our understanding of two and three dimensional dynamical systems, using in part some tools from low-dimensional topology. In these notes, I try to give a sense for how this works by discussing a few examples from problems I have worked on and surveying some related developments. Some symplectic variants of Weyl's law play a key role. I also briefly comment on some other developments and on the relationship with what is known in higher dimensions. The lectures themselves are available online.
This paper continues the study of link spectral invariants on compact surfaces, introduced in our previous work and shown to satisfy a Weyl law in which they asymptotically recover the Calabi invariant. Here we study their subleading asymptotics on surfaces of genus zero. We show the subleading asymptotics are bounded for smooth time-dependent Hamiltonians, and recover the Ruelle invariant for autonomous disc maps with finitely many critical values. We deduce that the Calabi homomorphism admits infinitely many extensions to the group of compactly supported area-preserving homeomorphisms, and that the kernel of the Calabi homomorphism on the group of hameomorphisms is not simple.
An influential result of McDuff and Schlenk asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase. This work has recently led to considerable interest in understanding when the ellipsoid embedding function for other symplectic 4-manifolds is partly described by an infinite staircase. We provide a general framework for analyzing this question for a large family of targets, called finite type convex toric domains, which we prove generalizes the class of closed toric symplectic 4-manifolds. When the target is of finite type, we prove that any infinite staircase must have a unique accumulation point a_0, given as the solution to an explicit quadratic equation. Moreover, we prove that the embedding function at a_0 must be equal to the classical volume lower bound. In particular, our result gives an obstruction to the existence of infinite staircases that we show is powerful. In the special case of rational convex toric domains, we can say more. We conjecture a complete answer to the question of existence of infinite staircases, in terms of six families that are distinguished by the fact that their moment polygon is reflexive. We then provide a uniform proof of the existence of infinite staircases for our six families, using two tools. For the first, we use recursive families of almost toric fibrations to find symplectic embeddings. For the second tool, we find recursive families of convex lattice paths that provide obstructions to embeddings. We conclude by reducing our conjecture that these are the only infinite staircases among rational convex toric domains to a question in number theory related to a classic work of Hardy and Littlewood.
We prove a compactness theorem for sequences of low-action punctured holomorphic curves of controlled topology, in any dimension, without imposing the typical assumption of uniformly bounded Hofer energy. In the limit, we extract a family of closed Reeb-invariant subsets. Then, we prove new structural results for the U-map in ECH and PFH, implying that such sequences exist in abundance in low-dimensional symplectic dynamics. We obtain applications to symplectic dynamics and the geometry of surfaces. First, we prove generalizations to higher genus surfaces and three-manifolds of the celebrated Le Calvez-Yoccoz theorem. Second, we show that for any closed Riemannian or Finsler surface a dense set of points have geodesics passing through them that visit different sections of the surface. Third, we prove a version of Ginzburg-Gürel's "crossing energy bound" for punctured holomorphic curves, of arbitrary topology, in symplectizations of any dimension.
We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, assuming that the associated contact structure has torsion first Chern class. As a special case, we prove a conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth and autonomous Hamiltonian flow on $\mathbb{R}^4$ has either two or infinitely many simple periodic orbits on any regular compact connected energy level that is transverse to the radial vector field. Other corollaries settle some old problems about Finsler metrics: we show that every Finsler metric on $S^2$ has either two or infinitely many prime closed geodesics; and we show that a Finsler metric on $S^2$ with at least one closed geodesic that is not irrationally elliptic must have infinitely many prime closed geodesics. The novelty of our work is that we do not make any nondegeneracy hypotheses.
We show that the space of open subsets of any complete and exact symplectic $4$-manifold has infinite dimension with respect to the symplectic Banach-Mazur distance; the quasi-flats we construct take values in the set of dynamically convex domains. In the case of $\mathbb{R}^4$, we therefore obtain the following contrast: the space of convex domains is quasi-isometric to a plane, while the space of dynamically convex ones has infinite dimension. In the case of $T^* S^2$, a variant of our construction resolves a conjecture of Stojisavljević and Zhang, asserting that the space of star-shaped domains in $T^* S^2$ has infinite dimension. Another corollary is that the space of contact forms giving the standard contact structure on $S^3$ has infinite dimension with respect to the contact Banach-Mazur distance.
A finite volume symplectic manifold is said to have "packing stability" if the only obstruction to symplectically embedding sufficiently small balls is the volume obstruction. Packing stability has been shown in a variety of cases and it has been conjectured that it always holds. We give counterexamples to this conjecture; in fact, we give examples that cannot be fully packed by any domain with smooth boundary nor by any convex domain. The examples are symplectomorphic to open and bounded domains in $\mathbb{R}^4$, with the diffeomorphism type of a disc. The obstruction to packing stability is closely tied to another old question, which asks to what extent an open symplectic manifold has a well-defined boundary; it follows from our results that many examples cannot be symplectomorphic to the interior of a compact symplectic manifold with smooth boundary. Our results can be quantified in terms of the volume decay near the boundary, and we produce, for example, smooth toric domains that are only symplectomorphic to the interior of a compact domain if the boundary of this domain has inner Minkowski dimension arbitrarily close to $4$. The growth rate of the subleading asymptotics of the ECH spectrum plays a key role in our arguments. We prove a very general "fractal Weyl law", relating this growth rate to the Minkowski dimension; this formula is potentially of independent interest.
A theorem of Gutt-Hutchings-Ramos asserts that all normalized symplectic capacities give the same value for monotone four-dimensional toric domains. We generalize this theorem to arbitrary dimension. The new ingredient in our proof is the construction of symplectic embeddings of "$L$-shaped" domains in any dimension into corresponding infinite cylinders; this resolves a conjecture of Gutt-Pereira-Ramos in the affirmative.
We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi invariant of Hamiltonians in the limit. We also prove a strong non-vanishing result, showing that under a monotonicity assumption which holds for a dense set of maps, the Periodic Floer Homology has infinite rank. An application of these results yields that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic points. This settles Smale's tenth problem in the special case of area-preserving diffeomorphisms of closed surfaces.
It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation; the contact struture is universally tight; and in the case of the three-sphere, the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid.
The third named author has been developing a theory of "higher" symplectic capacities. These capacities are invariant under taking products, and so are well-suited for studying the stabilized embedding problem. The aim of this note is to apply this theory, assuming its expected properties, to solve the stabilized embedding problem for integral ellipsoids, when the eccentricity of the domain has the opposite parity of the eccentricity of the target and the target is not a ball. For the other parity, the embedding we construct is definitely not always optimal; also, in the ball case, our methods recover previous results of McDuff, and of the second named author and Kerman. There is a similar story, with no condition on the eccentricity of the target, when the target is a polydisc: a special case of this implies a conjecture of the first named author, Frenkel, and Schlenk concerning the rescaled polydisc limit function. Some related aspects of the stabilized embedding problem and some open questions are also discussed.
Edtmair-Hutchings have recently defined, using periodic Floer homology, a U-cycle property for Hamiltonian isotopy classes of area-preserving diffeomorphisms of closed surfaces. They show that every Hamiltonian isotopy class satisfying the U-cycle property satisfies the smooth closing lemma and also satisfies a kind of Weyl law involving the actions of certain periodic points; they show that every rational isotopy class on the two-torus satisfies the U-cycle property. It seems that in general, not much is known about the U-module structure on PFH. Here we consider a version of Seiberg-Witten-Floer cohomology which is known by the work of Lee-Taubes to be isomorphic, as a U-module, to the periodic Floer homology in sufficiently high degree. We show that the analogous U-cycle property holds for every rational Hamiltonian isotopy class on any closed surface and, more generally, for any non-torsion spin-c structure. On the other hand, we also show that a rational isotopy class may contain elements that are not U-cyclic. By the Lee-Taubes isomorphism, the same results hold for PFH. Our results are some of the first computations concerning the U-module structure on these theories.
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 open disc, equipped with an area-form of finite area, is not perfect. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology.
In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for $n \ge 3$, asked if the same statement holds in dimension $2$. We show that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple. This settles what is known as the "simplicity conjecture" in the affirmative. In fact, we prove the a priori stronger statement that this group is not perfect. An important step in our proof involves verifying for certain smooth twist maps a conjecture of Hutchings concerning recovering the Calabi invariant from the asymptotics of spectral invariants defined using periodic Floer homology. Another key step, which builds on recent advances in continuous symplectic topology, involves proving that these spectral invariants extend continuously to area-preserving homeomorphisms of the disc. These two properties of PFH spectral invariants are potentially of independent interest. Our general strategy is partially inspired by suggestions of Fathi and the approach of Oh towards the simplicity question. In particular, we show that infinite twist maps, studied by Oh, are not finite energy homeomorphisms, which resolves the "infinite twist conjecture" in the affirmative; these twist maps are now the first examples of Hamiltonian homeomorphisms which can be said to have infinite energy. Another consequence of our work is that various forms of fragmentation for volume preserving homeomorphisms which hold for higher dimensional balls fail in dimension two.