arXiv subjects
Publications and source records attributed to Jakob Hedicke.
We show that the overtwisted contact structure on $S^3$ with Hopf invariant $-1$ is non-orderable, i.e., that it admits a contractible positive loop of contactopmorphisms. This provides the first known example of a non-orderable overtwisted contact manifold. We further show the existence of a contactomorphism without translated points.
We show that there is no universal upper bound for the systolic ratio of Bott-integrable contact forms on closed 3-manifolds, thus providing further evidence for the relative flexibility of integrable contact forms. For the proof, we study piecewise linear approximations of Lutz forms and establish integrability of a `plug' constructed by Abbondandolo, Bramham, Hryniewicz and Salomão for pushing up the systolic ratio.
Often it is possible to equip the space of all cone geodesics of a strongly convex cone structure with the structure of a smooth contact manifold. This generalizes the analogous notions for the space of light rays of a Lorentzian spacetime. After reviewing these constructions on the space of cone geodesics, with a focus on the natural contact structure, we establish a correspondence between positive paths of contactomorphisms in spherical cotangent bundles and certain globally hyperbolic cone structures.
Given a cooriented contact manifold $(M,ξ)$, it is possible to define a notion of positivity on the group $\mathrm{Diff}(M)$ of diffeomorphisms of $M$, by looking at paths of diffeomorphisms that are positively transverse to the contact distribution $ξ$. We show that, in contrast to the analogous notion usually considered on the group of diffeomorphisms preserving $ξ$, positivity on $\mathrm{Diff}(M)$ is completely flexible. In particular, we show that for the standard contact structure on $\mathbb{R}^{2n+1}$ any two diffeomorphisms are connected by a positive path. This result generalizes to compactly supported diffeomorphisms on a large class of contact manifolds. As an application we answer a question about Legendrians in thermodynamic phase space posed by Entov, Polterovich and Ryzhik in the context of thermodynamic processes.
We show that an overtwisted contact structure on a closed, oriented 3-manifold can be defined by a contact form having a Bott-integrable Reeb flow if and only if the Poincaré dual of its Euler class is represented by a graph link.
We relate non-orderability in contact topology to shortening in the contact Hofer norm. Combined with considerations of open books, this provides many new examples of non-orderable contact manifolds, including contact boundaries of subcritical Weinstein domains, and in particular the long-standing case of the standard $S^1 \times S^2.$ We also produce new examples of contact manifolds admitting contactomorphisms without translated points, provide obstructions to subcritical polarizations of symplectic manifolds, and establish a $\mathcal{C}^0$-continuity property of the contact Hofer metric.
An element $g$ of a Lie group is called stably elliptic if it is contained in the interior of the set $G^e$ of elliptic elements, characterized by the property that $\mathrm{Ad}(g)$ generates a relatively compact subgroup. Stably elliptic elements appear naturally in the geometry of causal symmetric spaces and in representation theory. We characterize stably elliptic elements in terms of the fixed point algebra of $\mathrm{Ad}(g)$ and show that the connected components of the set $G^{se}$ of stably elliptic elements can be described in terms of the Weyl group action on a compactly embedded Cartan subalgebra. In the case of simple hermitian Lie groups we relate stably elliptic elements to maximal invariant cones and the associated subsemigroups. In particular we show that the basic connected component $G^{se}(0)$ can be characterized in terms of the compactness of order intervals and that $G^{se}(0)$ is globally hyperbolic with respect to the induced biinvariant causal structure.
We show that there exist contact isotopies of the standard contact sphere whose time-1 maps do not have any translated points which are optimally close to the identity in the Shelukhin-Hofer distance. This proves the sharpness of a theorem of Shelukhin on the existence of translated points for contact isotopies of Liouville fillable contact manifolds with small enough Shelukhin-Hofer norm.
The symplectic cohomology of certain symplectic manifolds $W$ with non-compact ends modelled on the positive symplectization of a compact contact manifold $Y$ is shown to vanish whenever there is a positive loop of contactomorphisms of $Y$ which extends to a loop of Hamiltonian diffeomorphisms of $W$. An open string version of this result is also proved: the wrapped Floer cohomology of a Lagrangian $L$ with ideal Legendrian boundary $Λ$ is shown to vanish if there is a positive loop $Λ_{t}$ based at $Λ$ which extends to an exact loop of Lagrangians based at $L$. Various examples of such loops are considered. Applications include the construction of exotic compactly supported symplectomorphisms and exotic fillings of $Λ$.
This paper is devoted to studying a notion of Bott integrability for Reeb flows on contact 3-manifolds. We show, in analogy with work of Fomenko-Zieschang on Hamiltonian flows in dimension 4, that Bott-integrable Reeb flows exist precisely on graph manifolds. We also show that all $S^1$-invariant contact structures on Seifert manifolds, as well as all contact structures on the 3-sphere, on the 3-torus, and on $S^1\times S^2$, admit Bott-integrable Reeb flows. Along the way, we establish some general Liouville-type theorems for Bott-integrable Reeb flows, and a number of topological constructions (connected sum, open books, Dehn surgery) that may be expected to have wider applications.
We show that the natural conjugation invariant cone structure on the linear symplectic group $\mathrm{Sp}(2n)$ is globally hyperbolic in the positively elliptic region $\mathrm{Sp}_{\mathrm{ell}}^{+}(2n)$. This answers a question by Abbondandolo, Benedetti and Polterovich and shows a formula for a bi-invariant Lorentzian distance function dened by these authors for elements in this region. Moreover we give a characterisation of the positively elliptic region and of $-id \in\mathrm{Sp}(2n)$ in terms of the causality of this cone structure.
We define a Lorentzian distance function on the group of contactomorphisms of a closed contact manifold. This distance function is continuous with respect to the Hofer norm on the group of contactomorphisms defined by Shelukhin and finite if and only if the group of contactomorphisms is orderable. To prove this we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined by Rosen and Zhang is non-degenerate. In this case similar results hold for a Lorentzian distance function on Legendrian isotopy classes. This leads to a natural class of metrics associated to a globally hyperbolic Lorentzian manifold such that its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy class of the fibres.
We consider pseudoconvexity properties in Lorentzian and Riemannian manifolds and their relationship in static spacetimes. We provide an example of a causally continuous and maximal null pseudoconvex spacetime that fails to be causally simple. Its Riemannian factor provides an analogous example of a manifold that is minimally pseudoconvex, but fails to be convex.
It is shown that the space of null geodesics of a star-shaped causally simple subset of Minkowski space is contactomorphic to the canonical contact structure in the spherical cotangent bundle of $\mathbb{R}^n$. In the $3$-dimensional case we prove a similar result for a large class of causally simple contractible subsets of an arbitrary globally hyperbolic spacetime applying methods from the theory of contact-convex surfaces. Moreover we prove that under certain assumptions the space of null geodesics of a causally simple spacetime embeds with smooth boundary into the space of null geodesics of a globally hyperbolic spacetime. The characteristic foliation of this boundary provides an invariant of the conformal class of the causally simple spacetime.
It is shown that the space of null geodesics of a causally simple Lorentzian manifold is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions. Examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones.