arXiv subjects

Michael Hutchings

Publications and source records attributed to Michael Hutchings.

At least 19 recordsLinked to original sources

arXiv2026

Zeta functions of dynamically tame Liouville domains

We define a dynamical zeta function for nondegenerate Liouville domains, in terms of Reeb dynamics on the boundary. We use filtered equivariant symplectic homology to (i) extend the definition of the zeta function to a more general class of "dynamically tame" Liouville domains, and (ii) show that the zeta function of a dynamically tame Liouville domain is invariant under exact symplectomorphism of the interior. As an application, we find examples of open domains in R^4, arbitrarily close to a ball, which are not symplectomorphic to open star-shaped toric domains.

math.SG↗

arXiv2026

Elementary spectral invariants and three-dimensional Reeb dynamics

We survey various recent results on the existence and properties of periodic orbits of Reeb vector fields in three dimensions. We give an introduction to the "elementary spectral invariants" of contact three-manifolds, and we explain how they can be used to prove some of these results. (The remaining results can be proved using spectral invariants from embedded contact homology, of which the elementary spectral invariants are a simplification.) We then review the "alternative ECH capacities" of symplectic four-manifolds, and explain how these can be modified to define the elementary spectral invariants.

math.SG↗

arXiv2025

Anchored symplectic embeddings

Given two four-dimensional symplectic manifolds, together with knots in their boundaries, we define an ``anchored symplectic embedding'' to be a symplectic embedding, together with a two-dimensional symplectic cobordism between the knots (in the four-dimensional cobordism determined by the embedding). We use techniques from embedded contact homology to determine quantitative critera for when anchored symplectic embeddings exist, for many examples of toric domains. In particular we find examples where ordinarily symplectic embeddings exist, but they cannot be upgraded to anchored symplectic embeddings unless one enlarges the target domain.

math.SG↗

arXiv2025

Hopf orbits and the first ECH capacity

We consider dynamically convex star-shaped domains in a symplectic vector space of dimension $4$. For such a domain, a ``Hopf orbit'' is a closed characteristic in the boundary which is unknotted and has self-linking number $-1$. We show that the minimum action among Hopf orbits exists and defines a symplectic capacity for dynamically convex star-shaped domains. We further show that this capacity agrees with the first ECH capacity for such domains. Combined with a result of Edtmair, this implies that for dynamically convex star-shaped domains in four dimensions, the first ECH capacity agrees with the cylinder capacity. This also provides a method to show that the first ECH capacity of a dynamically convex star-shaped domain satisfies the axioms of a normalized symplectic capacity without any need for Seiberg-Witten theory.

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arXiv2025

Braid stability for periodic orbits of area-preserving surface diffeomorphisms

We consider an area-preserving diffeomorphism of a compact surface, which is assumed to be an irrational rotation near each boundary component. A finite set of periodic orbits of the diffeomorphism gives rise to a braid in the mapping torus. We show that under some nondegeneracy hypotheses, the isotopy classes of braids that arise from finite sets of periodic orbits are stable under Hamiltonian perturbations that are small with respect to the Hofer metric. A corollary is that for a Hamiltonian isotopy class of such maps, the topological entropy is lower semicontinuous with respect to the Hofer metric. This extends results of Alves-Meiwes for braids arising from finite sets of fixed points of Hamiltonian surface diffeomorphisms.

math.DS↗

arXiv2025

Reeb orbits frequently intersecting a symplectic surface

Consider a symplectic surface in a three-dimensional contact manifold with boundary on Reeb orbits (periodic orbits of the Reeb vector field). We assume that the rotation numbers of the boundary Reeb orbits satisfy a certain inequality, and we also make a technical assumption that the Reeb vector field has a particular ``nice'' form near the boundary of the surface. We then show that there exist Reeb orbits which intersect the interior of the surface, with a lower bound on the frequency of these intersections in terms of the symplectic area of the surface and the contact volume of the three-manifold. No genericity of the contact form is assumed. As a corollary of the main result, we obtain a generalization of various recent results relating the mean action of periodic orbits to the Calabi invariant for area-preserving surface diffeomorphisms.

math.SG↗

arXiv2024

Elementary spectral invariants and quantitative closing lemmas for contact three-manifolds

In a previous paper, we defined an "elementary" alternative to the ECH capacities of symplectic four-manifolds, using max-min energy of holomorphic curves subject to point constraints, and avoiding the use of Seiberg-Witten theory. In the present paper we use a variant of this construction to define an alternative to the ECH spectrum of a contact three-manifold. The alternative spectrum has applications to Reeb dynamics in three dimensions. In particular, we adapt ideas from a previous joint paper with Edtmair to obtain quantitative closing lemmas for Reeb vector fields in three dimensions. For the example of an irrational ellipsoid, we obtain a sharp quantitative closing lemma.

math.SG↗

arXiv2024

PFH spectral invariants and $C^\infty$ closing lemmas

We develop the theory of spectral invariants in periodic Floer homology (PFH) of area-preserving surface diffeomorphisms. We use this theory to prove $C^\infty$ closing lemmas for certain Hamiltonian isotopy classes of area-preserving surface diffeomorphisms. In particular, we show that for a $C^\infty$-generic area-preserving diffeomorphism of the torus, the set of periodic points is dense. Our closing lemmas are quantitative, asserting roughly speaking that for a given Hamiltonian isotopy, within time $δ$ a periodic orbit must appear of period $O(δ^{-1})$. We also prove a "Weyl law" describing the asymptotic behavior of PFH spectral invariants.

math.SG↗

arXiv2024

Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture

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.

math.SG↗

arXiv2022

An elementary alternative to ECH capacities

The ECH capacities are a sequence of numerical invariants of symplectic four-manifolds which give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg-Witten theory. In this note we define a new sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The new capacities satisfy the same basic properties as ECH capacities and agree with the ECH capacities for the main examples for which the latter have been computed, namely convex and concave toric domains. The new capacities are also useful for obstructing symplectic embeddings into closed symplectic four-manifolds. This work is inspired by a recent preprint of McDuff-Siegel giving a similar elementary alternative to symplectic capacities from rational SFT.

math.SG↗

arXiv2022

S^1-equivariant contact homology for hypertight contact forms

In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define "nonequivariant contact homology" and "S^1-equivariant contact homology", both with integer coefficients, for a contact form on a closed manifold in any dimension with no contractible Reeb orbits. We prove that these contact homologies depend only on the contact structure. Our construction uses Morse-Bott theory and is related to the positive S^1-equivariant symplectic homology of Bourgeois-Oancea. However, instead of working with Hamiltonian Floer homology, we work directly in contact geometry, using families of almost complex structures. When cylindrical contact homology can also be defined, it agrees with the tensor product of the S^1-equivariant contact homology with Q. We also present examples showing that the S^1-equivariant contact homology contains interesting torsion information. In a subsequent paper we will use obstruction bundle gluing to extend the above story to closed three-manifolds with dynamically convex contact forms, which in particular will prove that their cylindrical contact homology has a lift to integer coefficients which depends only on the contact structure.

math.SG↗

arXiv2022

Contact three-manifolds with exactly two simple Reeb orbits

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.

math.SG↗

arXiv2022

ECH capacities and the Ruelle invariant

The ECH capacities are a sequence of real numbers associated to any symplectic four-manifold, which are monotone with respect to symplectic embeddings. It is known that for a compact star-shaped domain in R^4, the ECH capacities asymptotically recover the volume of the domain. We conjecture, with a heuristic argument, that generically the error term in this asymptotic formula converges to a constant determined by a "Ruelle invariant" which measures the average rotation of the Reeb flow on the boundary. Our main result is a proof of this conjecture for a large class of toric domains. As a corollary, we obtain a general obstruction to symplectic embeddings of open toric domains with the same volume. For more general domains in R^4, we bound the error term with an improvement on the previously known exponent from 2/5 to 1/4.

math.SG↗

arXiv2021

Computing Reeb dynamics on 4d convex polytopes

We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio $1$.

math.SG↗

arXiv2020

Examples around the strong Viterbo conjecture

A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on $S^1$-invariant convex domains. We introduce a new class of examples called "monotone toric domains", which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capacities agree. For monotone toric domains in arbitrary dimension, we prove that the Gromov width agrees with the first equivariant capacity. We also study a family of examples of non-monotone toric domains and determine when the conclusion of the strong Viterbo conjecture holds for these examples. Along the way we compute the cylindrical capacity of a large class of "weakly convex toric domains" in four dimensions.

math.SG↗

arXiv2019

Axiomatic S^1 Morse-Bott theory

In various situations in Floer theory, one extracts homological invariants from "Morse-Bott" data in which the "critical set" is a union of manifolds, and the moduli spaces of "flow lines" have evaluation maps taking values in the critical set. This requires a mix of analytic arguments (establishing properties of the moduli spaces and evaluation maps) and formal arguments (defining or computing invariants from the analytic data). The goal of this paper is to isolate the formal arguments, in the case when the critical set is a union of circles. Namely, we state axioms for moduli spaces and evaluation maps (encoding a minimal amount of analytical information that one needs to verify in any given Floer-theoretic situation), and using these axioms we define homological invariants. More precisely, we define a (almost) category of "Morse-Bott systems". We construct a "cascade homology" functor on this category, based on ideas of Bourgeois and Frauenfelder, which is "homotopy invariant". This machinery is used in our work on cylindrical contact homology.

math.SG↗

arXiv2018

Torsion contact forms in three dimensions have two or infinitely many Reeb orbits

We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it follows that under the above assumptions, there are infinitely many simple Reeb orbits if the three-manifold is not the three-sphere or a lens space. We also show that for non-torsion contact structures, every nondegenerate contact form has at least four simple Reeb orbits.

math.SG↗

arXiv2017

Symplectic capacities from positive S^1-equivariant symplectic homology

We use positive S^1-equivariant symplectic homology to define a sequence of symplectic capacities c_k for star-shaped domains in R^{2n}. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities c_k of any "convex toric domain" or "concave toric domain". As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities c_k to functions of Liouville domains which are almost but not quite symplectic capacities.

math.SG↗