Characterizing the open contact manifolds that satisfy the Weinstein conjecture

The Weinstein conjecture predicts that the Reeb vector field of every contact form defining a contact structure on a closed manifold has a periodic orbit; on open (non-compact) contact manifolds this statement fails outright — the standard contact structure on $\mathbb{R}^{2n+1}$ carries the Reeb field $\partial_z$, which has no periodic orbits. The problem, developing the discussion in Dörner–Geiges–Zehmisch's survey of periodic Reeb orbits — which defines property $(W)$ for contact manifolds, records simple open manifolds where $(W)$ fails together with recent progress establishing it for a number of open contact manifolds, and asks whether local changes in the topology or the contact form on $(\mathbb{R}^{2n+1},\alpha_{st})$ force periodic Reeb orbits — is to characterize the open contact manifolds $(M,\xi)$ for which every defining contact form has a periodic Reeb orbit, and in the model case of contact forms on $(\mathbb{R}^{2n+1},\xi_{st})$ that are standard at infinity, to find dynamical conditions forcing periodic orbits in dimensions $2n+1\ge 5$, where Geiges–Röttgen–Zehmisch showed that even the existence of trapped orbits does not suffice. Despite sustained activity (virtually contact structures, cotangent bundles over non-compact bases, and the singular $b^m$-contact line), no general characterization and no higher-dimensional analogue of the Eliashberg–Hofer trapped-orbit theorem is known.

Background

A contact form on a $(2n+1)$-dimensional manifold $M$ is a 1-form $\alpha$ such that $\alpha\wedge(d\alpha)^n$ is nowhere vanishing; its Reeb vector field $R_\alpha$ is uniquely determined by $\iota_{R_\alpha}d\alpha=0$ and $\alpha(R_\alpha)=1$. The hyperplane field $\xi=\ker\alpha$ is a (cooriented) contact structure, and $\alpha$ is unique up to multiplication by a positive function, so properties of the Reeb flow that hold for every defining form are properties of the contact structure $(M,\xi)$ itself. A periodic Reeb orbit is a trajectory that closes up; it is contractible if it is null-homotopic in $M$ (the period need not be minimal). Weinstein (1979) conjectured that on a closed manifold the Reeb flow of every defining form has a periodic orbit — property $(W)$; a stronger property $(W^\circ)$ demands a contractible periodic orbit. For closed manifolds, $(W)$ is known in many cases, notably for all closed orientable 3-manifolds by Taubes's Seiberg–Witten argument, but remains open in full generality.

On open (non-compact) manifolds the naive conjecture fails: for the standard contact form $\alpha_{st}=dz+\frac{1}{2}\sum_j(x_j\,dy_j-y_j\,dx_j)$ on $\mathbb{R}^{2n+1}$, the Reeb field is $\partial_z$, which has no periodic orbits. The meaningful question is therefore which open contact manifolds satisfy $(W)$. A contact form is called standard at infinity if it agrees with $\alpha_{st}$ outside a compact set, and a Reeb orbit is trapped if it enters a compact region without leaving it again (but never closes up). Eliashberg and Hofer (1994), via filling arguments with holomorphic discs (discs solving a Cauchy–Riemann type equation in the symplectization, the cylinder $\mathbb{R}\times M$ equipped with the symplectic form $d(e^t\alpha)$), proved a global Darboux theorem: a contact form on $\mathbb{R}^3$ that is standard at infinity and has no periodic Reeb orbit is diffeomorphic to the standard pair; consequently, in dimension three a trapped orbit forces a periodic one. This picture collapses in higher dimensions: Geiges–Röttgen–Zehmisch (Invent. Math. 2014) constructed, for every $n\ge 2$, a contact form $\alpha=h\,\alpha_{st}$ with $h>0$ and $h-1$ compactly supported — the contact structure $\xi_{st}$ is unchanged — whose Reeb flow has a trapped orbit (built from an irrational linear flow on a Clifford torus, a product of circles in $\mathbb{C}^n$) but no periodic orbit at all. Related work sharpens both sides: Reeb dynamics detects the diffeomorphism type of odd-dimensional balls (Geiges–Zehmisch), Reeb traps can be converted into Hamiltonian plugs (Geiges–Röttgen–Zehmisch), and minimal sets of Denjoy type — aperiodic minimal invariant sets borrowed from Denjoy's classical construction in circle dynamics — can trap Reeb orbits (Arai–Inaba–Kano). On the positive side, periodic orbits are forced on special classes of open contact manifolds: starshaped hypersurfaces (bounding a region star-shaped with respect to a point) and non-compact hypersurfaces in $\mathbb{R}^{2n}$, cotangent bundles over non-compact bases (van den Berg–Pasquotto–Vandervorst and coauthors), via linking of orbits with compact sets (Suhr–Zehmisch), and on non-compact covers of closed manifolds carrying a virtually contact structure — a lift of a contact form to a cover that is bounded with all derivatives (Bae 2014; Bae–Wiegand–Zehmisch 2020). More recently, the singular $b^m$-contact setting (contact forms singular along a hypersurface, studied by Miranda–Oms and Peralta-Salas) was shown to run hand-in-hand with the Weinstein conjecture on non-compact manifolds with convex ends, where again only partial existence results are available and counterexamples to the naive singular conjecture exist.

Problem Statement

Let $(M,\xi)$ be an open (non-compact) cooriented contact manifold, and say that $(M,\xi)$ satisfies the Weinstein conjecture, property $(W)$, if the Reeb vector field of every contact form defining $\xi$ has a periodic orbit (the contractible-orbit strengthening $(W^\circ)$ may be treated in parallel).

(i) Characterize the open contact manifolds satisfying $(W)$: give necessary and sufficient conditions — in terms of the topology of $M$, invariants of the contact structure $\xi$, or dynamical properties of defining forms — under which periodic Reeb orbits are forced for every defining contact form. Since $(W)$ is by definition a property of the pair $(M,\xi)$, an admissible characterization is a criterion that decides, for each open contact manifold, whether it satisfies $(W)$, in terms of data other than the periodic-orbit property itself; a condition that merely restates $(W)$ does not qualify — for instance 'there exists a defining form $\alpha$ of $\xi$ such that every defining form of $\xi$ has a periodic Reeb orbit', which is equivalent to $(W)$ only because the class of defining forms is nonempty. Conditions may be phrased through the topology of $M$, through invariants of $\xi$, or through properties of defining forms, provided they are not reformulations of $(W)$ (or of $(W^\circ)$).

(ii) Model case: on $(\mathbb{R}^{2n+1},\ker\alpha_{st})$ with $n\ge 2$, consider contact forms $\alpha=h\,\alpha_{st}$ with $h>0$ and $h-1$ compactly supported, i.e. contact forms standard at infinity. Since Geiges–Röttgen–Zehmisch showed that the existence of a trapped orbit does not force a periodic orbit in these dimensions, identify dynamical conditions on the Reeb flow $R_\alpha$ — in the spirit of the Eliashberg–Hofer theorem for $n=1$ — that do force a periodic Reeb orbit. Such a condition must be checkable from the Reeb dynamics without presupposing the existence of a periodic orbit: the tautological condition '$R_\alpha$ has a periodic orbit' is excluded, since it is trivially sufficient and its failure is trivially aperiodic behavior.

Parts (i) and (ii) address one scientific objective: determining exactly when periodic Reeb orbits are forced on open contact manifolds. They are not interchangeable, however. $(\mathbb{R}^{2n+1},\xi_{st})$ itself fails $(W)$, so (ii) is a question about individual contact forms standard at infinity, whereas (i) asks for a property of open contact manifolds that holds for all their defining forms; a result of type (ii) therefore does not by itself characterize the open contact manifolds satisfying $(W)$. Both parts are required: a resolution must supply a characterization for (i) and at least one non-tautological forcing condition for (ii), so a complete answer to (i) alone does not resolve the task.

References

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