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Will J. Merry

Publications and source records attributed to Will J. Merry.

17 recordsLinked to original sources
arXiv2022

The symplectic cohomology of magnetic cotangent bundles

We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles, without any restrictions on the magnetic form, using the dissipative method for compactness introduced in \cite{Groman2015}. As an application, we deduce that if $N$ is a closed manifold and $ σ$ is a magnetic form that is not weakly exact, then the $ π_1$-sensitive Hofer-Zehnder capacity of any compact set in the magnetic cotangent bundle determined by $ σ$ is finite.

math.SG↗
arXiv2020

Computing the Rabinowitz Floer homology of tentacular hyperboloids

We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids $Σ\simeq S^{n+k-1}\times\mathbb{R}^{n-k}$. Using an embedding of a compact sphere $Σ_0\simeq S^{2k-1}$ into the hypersurface $Σ$, we construct a chain map from the Floer complex of $Σ$ to the Floer complex of $Σ_0$. In contrast to the compact case, the Rabinowitz Floer homology groups of $Σ$ are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.

math.SG↗
arXiv2017

Positive loops and $L^{\infty}$-contact systolic inequalities

We prove an inequality between the $L^{\infty}$-norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds was proved by Casals-Presas-Sandon in [CPS16]. As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimizers of the $L^\infty$-norm. Moreover, we establish $L^\infty$-type contact systolic inequalities in the presence of a positive loop.

math.SG↗
arXiv2017

Maximum principles in symplectic homology

In the setting of symplectic manifolds which are convex at infinity, we use a version of the Aleksandrov maximum principle to derive uniform estimates for Floer solutions that are valid for a wider class of Hamiltonians and almost complex structures than is usually considered. This allows us to extend the class of Hamiltonians which one can use in the direct limit when constructing symplectic homology. As an application, we detect elements of infinite order in the symplectic mapping class group of a Liouville domain, and obtain existence results for translated points.

math.SG↗
arXiv2014

On the existence of infinitely many invariant Reeb orbits

In this article we extend results of Grove and Tanaka on the existence of isometry-invariant geodesics to the setting of Reeb flows and strict contactomorphisms. Specifically, we prove that if M is a closed connected manifold with the property that the Betti numbers of the free loop space are asymptotically unbounded then for every fibrewise star-shaped hypersurface in the cotangent bundle of M and every strict contactomorphism of that hypersurface which is contact-isotopic to the identity, there are infinitely many invariant Reeb orbits.

math.SG↗
arXiv2014

Orderability, contact non-squeezing, and Rabinowitz Floer homology

We study Liouville fillable contact manifolds $(Σ,ξ)$ with non-zero Rabinowitz Floer homology and assign spectral numbers to paths of contactomorphisms. As a consequence we prove that $\widetilde{\mathrm{Cont}_0}(Σ,ξ)$ is orderable in the sense of Eliashberg and Polterovich. This provides a new class of orderable contact manifolds. If the contact manifold is in addition periodic or a prequantization space $M \times S^1$ for $M$ a Liouville manifold, then we construct a contact capacity. This can be used to prove a general non-squeezing result, which amongst other examples in particular recovers the beautiful non-squeezing results from [EKP06].

math.SG↗
arXiv2014

Orderability and the Weinstein Conjecture

In this article we prove that the Weinstein conjecture holds for contact manifolds $(Σ,ξ)$ for which $\mathrm{Cont}_0(Σ,ξ)$ is non-orderable in the sense of Eliashberg-Polterovich [EP00]. More precisely, we establish a link between orderable and hypertight contact manifolds. In addition, we prove for certain contact manifolds a conjecture by Sandon [San13b] on the existence of translated points in the non-degenerate case.

math.SG↗
arXiv2013

Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

We study the following rigidity problem in symplectic geometry:can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser. Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the `virtually contact' setting. By means of an Abbondandolo-Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.

math.SG↗
arXiv2013

Floer homology for non-resonant magnetic fields on flat tori

In this article we define and compute the Novikov Floer homology associated to a non-resonant magnetic field and a mechanical Hamiltonian on a flat torus T^{2N}. As a result, we deduce that this Hamiltonian system always has 2N+1 contractible solutions, and generically even 2^{2N} contractible solutions. Moreover if there exists a non-degenerate non-contractible solution then there necessarily exists another.

math.SG↗
arXiv2013

Translated points and Rabinowitz Floer homology

We prove that if a contact manifold admits an exact filling then every local contactomorphism isotopic to the identity admits a translated point in the interior of its support, in the sense of Sandon [San11b]. In addition we prove that if the Rabinowitz Floer homology of the filling is non-zero then every contactomorphism isotopic to the identity admits a translated point, and if the Rabinowitz Floer homology of the filling is infinite dimensional then every contactmorphism isotopic to the identity has either infinitely many translated points, or a translated point on a closed leaf. Moreover if the contact manifold has dimension greater than or equal to 3, the latter option generically doesn't happen. Finally, we prove that a generic contactomorphism on $\mathbb{R}^{2n+1}$ has infinitely many geometrically distinct iterated translated points all of which lie in the interior of its support.

math.SG↗
arXiv2012

Floer homology for magnetic fields with at most linear growth on the universal cover

The Floer homology of a cotangent bundle is isomorphic to loop space homology of the underlying manifold, as proved by Abbondandolo-Schwarz, Salamon-Weber, and Viterbo. In this paper we show that in the presence of a Dirac magnetic monopole which admits a primitive with sublinear growth on the universal cover, the Floer homology in atoroidal free homotopy classes is again isomorphic to loop space homology. As a consequence we prove that for any atoroidal free homotopy class and any sufficiently small T>0, any magnetic flow associated to the Dirac magnetic monopole has a closed orbit of period T belonging to the given free homotopy class. In the case where the Dirac magnetic monopole admits a bounded primitive on the universal cover we also prove the Conley conjecture for Hamiltonians that are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.

math.SG↗
arXiv2011

On the growth rate of leaf-wise intersections

We define a new variant of Rabinowitz Floer homology that is particularly well suited to studying the growth rate of leaf-wise intersections. We prove that for closed manifolds $M$ whose loop space is "complicated", if $Σ$ is a non-degenerate fibrewise starshaped hypersurface in $T^*M$ and $ϕ$ is a generic Hamiltonian diffeomorphism then the number of leaf-wise intersection points of $ϕ$ in $Σ$ grows exponentially in time. Concrete examples of such manifolds $M$ are the connected sum of two copies of $S^2 \times S^2$, the connected sum of $T^4$ and $CP^2$, or any surface of genus greater than one.

math.SG↗
arXiv2010

On the Rabinowitz Floer homology of twisted cotangent bundles

Consider the cotangent bundle of a Riemannian manifold $(M,g)$ of dimension 2 or more, endowed with a twisted symplectic structure defined by a closed weakly exact 2-form $σ$ on $M$ whose lift to the universal cover of $M$ admits a bounded primitive. We compute the Rabinowitz Floer homology of energy hypersurfaces $Σ_{k}=H^{-1}(k)$ of mechanical (kinetic energy + potential) Hamiltonians $H$ for the case when the energy value k is greater than the Mane critical value c. Under the stronger condition that k>c_{0}, where c_{0} denotes the strict Mane critical value, Abbondandolo and Schwarz recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k>c, thus covering cases where $σ$ is not exact. As a consequence, we deduce that the hypersurface corresponding to the energy level k is never displaceable for any k>c. Moreover, we prove that if dim M > 1, the homology of the free loop space of $M$ is infinite dimensional, and if the metric is chosen generically, a generic Hamiltonian diffeomorphism has infinitely many leaf-wise intersection points in $Σ_{k}$.

math.SG↗
arXiv2010

Index computations in Rabinowitz Floer homology

In this note we study two index questions. In the first we establish the relationship between the Morse indices of the free time action functional and the fixed time action functional. The second is related to Rabinowitz Floer homology. Our index computations are based on a correction term which is defined as follows: around a non-degenerate Hamiltonian orbit lying in a fixed energy level a well-known theorem says that one can find a whole cylinder of orbits parametrized by the energy. The correction term is determined by whether the periods of the orbits are increasing or decreasing as one moves up the orbit cylinder. We also provide an example to show that, even above the Mañé critical value, the periods may be increasing thus producing a jump in the Morse index of the free time action functional in relation to the Morse index of the fixed time action functional.

math.SG↗
arXiv2010

Stability of Anosov Hamiltonian Structures

Consider the tangent bundle of a Riemannian manifold $(M,g)$ of dimension $n\geq3$ admitting a metric of negative curvature (not necessarily equal to $g$) endowed with a twisted symplectic structure defined by a closed 2-form on $M$. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by the standard kinetic energy Hamiltonian, and we consider a compact regular energy level $Σ_{k}:=H^{-1}(k)$ of $H$. Suppose $Σ_{k}$ is an Anosov energy level. We prove that if $n$ is odd, then if the Hamiltonian flow restricted to $Σ_{k}$ is Anosov with $C^{1}$ weak bundles then the Hamiltonian structure $(Σ_{k}$ is stable if and only if it is contact. If $n$ is even and in addition the flow is assumed to be 1/2-pinched then the same conclusion holds. As a corollary we deduce that if $g$ is negatively curved, strictly 1/4-pinched and the 2-form defining the twisted symplectic structure is not exact then the Hamiltonian structure $(Σ_{k}$ is never stable for all sufficiently large $k$.

math.DS↗
arXiv2009

Closed orbits of a charge in a weakly exact magnetic field

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected Riemannian manifold and $σ$ a weakly exact 2-form. Let $ϕ_{t}$ denote the magnetic flow determined by $σ$, and let $c$ denote the Mane critical value of the pair $(g,σ)$. We prove that if $k>c$, then for every non-trivial free homotopy class of loops on $M$ there exists a closed orbit with energy $k$ whose projection to $M$ belongs to that free homotopy class. We also prove that for almost all $k<c$ there exists a closed orbit with energy $k$ whose projection to $M$ is contractible. In particular, when $c=\infty$ this implies that almost every energy level has a contractible closed orbit. As a corollary we deduce that if $σ$ is not exact and $M$ has an amenable fundamental group (which implies $c=\infty$) then there exist contractible closed orbits on almost every energy level.

math.DS↗