1.1 Recovery

We begin by recovering Theorem 1.1 in a different but equivalent setting. Let $\unicode[STIX]{x1D706}$ be a contact form on $M^{2n-1}$ . It defines a unique Reeb vector field $R_{\unicode[STIX]{x1D706}}$ on $M$ via the equations

$$\begin{eqnarray}i_{R_{\unicode[STIX]{x1D706}}}d\unicode[STIX]{x1D706}=0\quad \text{and}\quad \unicode[STIX]{x1D706}(R_{\unicode[STIX]{x1D706}})=1.\end{eqnarray}$$

It also defines a contact structure $\unicode[STIX]{x1D709}=\ker (\unicode[STIX]{x1D706})$ . Any other contact form defining the same contact structure is of the form $f\unicode[STIX]{x1D706}$ for some nonvanishing smooth function $f:M\rightarrow \mathbb{R}$ .

Let $\unicode[STIX]{x1D706}_{0}$ be the standard contact form on the unit sphere $\mathbb{S}^{2n-1}\subset \mathbb{R}^{2n}$ obtained by restricting the form $\frac{1}{2}\sum _{j=1}^{n}(p_{j}dq_{j}-q_{j}dp_{j})$ . The following result is equivalent to Theorem 1.1 for the case of smooth hypersurfaces.

Theorem 1.3. Let $\unicode[STIX]{x1D706}=f\unicode[STIX]{x1D706}_{0}$ for some positive function $f$ . If

(1) $$\begin{eqnarray}\frac{\max (f)}{\min (f)}<2,\end{eqnarray}$$

then there are at least $n$ distinct closed orbits of $R_{\unicode[STIX]{x1D706}}$ with periods in the interval $[\unicode[STIX]{x1D70B}\min (f),\unicode[STIX]{x1D70B}\max (f)].$

Recall that a closed orbit $\unicode[STIX]{x1D6FE}$ of $R_{\unicode[STIX]{x1D706}}$ (closed Reeb orbit of $\unicode[STIX]{x1D706}$ ) is said to be distinct from another such orbit $\widetilde{\unicode[STIX]{x1D6FE}}$ if for all $k\in \mathbb{N}$ and $c\in \mathbb{R}$ there is a $t\in \mathbb{R}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}(t)\neq \widetilde{\unicode[STIX]{x1D6FE}}(kt+c).\end{eqnarray}$$

Two closed orbits are said to be geometrically distinct if they have disjoint images. While geometrically distinct orbits are distinct, the converse does not hold since two distinct orbits can both be relatively prime multiples of a third closed orbit.

To see that Theorem 1.3 implies Theorem 1.1, consider a smooth hypersurface $\unicode[STIX]{x1D6F4}$ as in the statement of Theorem 1.1. It can be described in the form

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}=\{z\sqrt{f(z)}\mid z\in \mathbb{S}^{2n-1}\},\end{eqnarray}$$

where $f$ is a positive smooth function on $\mathbb{S}^{2n-1}$ . There is a bijective correspondence between the simple closed Reeb orbits of $\unicode[STIX]{x1D706}=f\unicode[STIX]{x1D706}_{0}$ with period $T$ (modulo translation) and the closed characteristics on $\unicode[STIX]{x1D6F4}$ with action equal to $T$ . To show that Theorem 1.1 follows from Theorem 1.3, it therefore suffices to show that the convexity of $\unicode[STIX]{x1D6F4}$ implies that the closed Reeb orbits of $\unicode[STIX]{x1D706}$ with period in $[\unicode[STIX]{x1D70B}\min (f),\unicode[STIX]{x1D70B}\max (f)]$ are simple and hence geometrically distinct. Since $\max (f)<2\min (f)$ , it suffices to show that the convexity of $\unicode[STIX]{x1D6F4}$ implies that $R_{\unicode[STIX]{x1D706}}$ has no closed orbits of period less than $\unicode[STIX]{x1D70B}\min (f)$ or, equivalently, that $\unicode[STIX]{x1D6F4}$ has no closed characteristics with action less than $\unicode[STIX]{x1D70B}\min (f)$ . This last condition follows immediately from the main result of Croke and Weinstein in [Reference Croke and WeinsteinCW81], which asserts that the closed characteristics of convex hypersurfaces containing the ball of radius $r$ have action at least $\unicode[STIX]{x1D70B}r^{2}$ .

Remark 1.5. Given a contact structure $\unicode[STIX]{x1D709}$ on $M$ , there is a natural coarse pseudo-metric on the space of contact forms defining $\unicode[STIX]{x1D709}$ . For any two such contact forms, say $\unicode[STIX]{x1D706}$ and $\tilde{\unicode[STIX]{x1D706}}$ , there is a smooth nonvanishing function $f_{\unicode[STIX]{x1D706}/\tilde{\unicode[STIX]{x1D706}}}$ on $M$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D706}=f_{\unicode[STIX]{x1D706}/\tilde{\unicode[STIX]{x1D706}}}\tilde{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

The distance between them can then be defined as

(2) $$\begin{eqnarray}d(\unicode[STIX]{x1D706},\tilde{\unicode[STIX]{x1D706}})=\ln \biggl(\frac{\max (f_{\unicode[STIX]{x1D706}/\tilde{\unicode[STIX]{x1D706}}})}{\min (f_{\unicode[STIX]{x1D706}/\tilde{\unicode[STIX]{x1D706}}})}\biggr).\end{eqnarray}$$

It is easy to verify that $d$ is a pseudo-metric and that its degeneracy only reflects the fact that it is invariant under scalar multiplication of the contact forms. In these terms, condition (1) can be rephrased geometrically as $d(\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}_{0})<\ln 2.$

1.2 Prequantization spaces

Our first generalization of Theorem 1.3 establishes the same rigidity phenomenon for all prequantization spaces. Consider a closed symplectic manifold $(Q,\unicode[STIX]{x1D714})$ such that the de Rham cohomology class $-[\unicode[STIX]{x1D714}]/2\unicode[STIX]{x1D70B}$ is the image of an integral class $\mathbf{e}\in H^{2}(Q;\mathbb{Z})$ . Let $p:M\rightarrow Q$ be an $\mathbb{S}^{1}$ -bundle over $Q$ with first Chern class equal to $\mathbf{e}$ . Denote the corresponding Boothby–Wang contact form on $M$ by $\unicode[STIX]{x1D706}_{Q}$ and the corresponding contact structure by $\unicode[STIX]{x1D709}_{Q}$ . We then have $d\unicode[STIX]{x1D706}_{Q}=p^{\ast }\unicode[STIX]{x1D714}$ and the Reeb vector field of $\unicode[STIX]{x1D706}_{Q}$ generates the circle action on $M$ with period $2\unicode[STIX]{x1D70B}$ .

Let $\unicode[STIX]{x1D6FC}_{\mathbf{f}}\in [\mathbb{S}^{1},M]$ be the free homotopy class corresponding to the fibres of the bundle $p:M\rightarrow Q$ and denote its order by $|\unicode[STIX]{x1D6FC}_{\mathbf{f}}|$ .

Theorem 1.6. Let $\unicode[STIX]{x1D706}=f\unicode[STIX]{x1D706}_{Q}$ for some positive function $f$ . If

$$\begin{eqnarray}\frac{\max (f)}{\min (f)}<2,\end{eqnarray}$$

then there are at least $\frac{1}{2}\dim (Q)+1$ distinct closed Reeb orbits of $\unicode[STIX]{x1D706}$ which represent the class $\unicode[STIX]{x1D6FC}_{\mathbf{f}}$ and have period in the interval

$$\begin{eqnarray}[2\unicode[STIX]{x1D70B}\min (f),2\unicode[STIX]{x1D70B}\max (f)].\end{eqnarray}$$

These orbits are geometrically distinct from one another if the class $\unicode[STIX]{x1D6FC}_{\mathbf{f}}$ is either primitive or of infinite order. Otherwise, they are geometrically distinct if there are no closed Reeb orbits of $\unicode[STIX]{x1D706}$ which have period less than or equal to

$$\begin{eqnarray}\frac{2\unicode[STIX]{x1D70B}}{|\unicode[STIX]{x1D6FC}_{\mathbf{f}}|}(\max (f)-\min (f))\end{eqnarray}$$

and which represent a class $\unicode[STIX]{x1D6FD}$ such that $\unicode[STIX]{x1D6FD}^{k}=\unicode[STIX]{x1D6FC}_{\mathbf{f}}$ for some integer $k>1$ .

Note that we can detect geometrically distinct orbits in many cases without a geometric assumption like convexity or an equivalent substitute like dynamical convexity.

1.3 Rigid constellations

We now present a more extensive generalization of the Ekeland–Lasry rigidity phenomenon. Let $(M,\unicode[STIX]{x1D706}_{0})$ be any closed contact manifold and let $\unicode[STIX]{x1D6FC}\in [\mathbb{S}^{1},M]$ be any free homotopy class. (The trivial class will be denoted here by $e$ .) We first identify collections of closed orbits of $\unicode[STIX]{x1D706}_{0}$ in class $\unicode[STIX]{x1D6FC}$ which can meaningfully influence the Reeb flows of contact forms which are nearby in the sense of Remark 1.5. As the original proof of Theorem 1.1 suggests, these collections should consist of simple orbits (so that the natural $\mathbb{R}/\mathbb{Z}$ -action is free) and their periods should be isolated in the period spectrum.

Let ${\mathcal{T}}(\unicode[STIX]{x1D706}_{0})$ be the set of periods of all closed Reeb orbits of $\unicode[STIX]{x1D706}_{0}$ and, assuming that it is nonempty, let $T_{\min }(\unicode[STIX]{x1D706}_{0})$ be the smallest such period. Restricting to orbits in class $\unicode[STIX]{x1D6FC}$ , we get the similarly defined set ${\mathcal{T}}(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})$ and minimal $\unicode[STIX]{x1D6FC}$ -period, $T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})$ .

Given a $T$ in ${\mathcal{T}}(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})$ , let ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ be the collection of closed orbits of $R_{\unicode[STIX]{x1D706}_{0}}$ which represent the class $\unicode[STIX]{x1D6FC}$ and have period in the interval $[T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}),T]$ .

Definition 1.8. The collection of closed orbits ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ is a rigid constellation if every orbit in ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ is simple, no decreasing sequence in ${\mathcal{T}}(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})$ converges to $T$ and

(3) $$\begin{eqnarray}T<T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})+T_{\min }(\unicode[STIX]{x1D706}_{0}).\end{eqnarray}$$

Given a rigid constellation ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ , we set

$$\begin{eqnarray}T^{+}=\min \{T^{\prime }\in {\mathcal{T}}(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}):T^{\prime }>T\}.\end{eqnarray}$$

This number is strictly greater than $T$ by the second condition of the definition above.

Since Reeb vector fields are autonomous, the elements of a rigid constellation ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ can be divided into separate $\mathbb{R}/\mathbb{Z}$ -families of closed Reeb orbits, the elements of which differ only by simple translation reparameterizations. It follows from the simplicity condition in Definition 1.8 that closed orbits belonging to different $\mathbb{R}/\mathbb{Z}$ -families of a rigid constellation are geometrically distinct from one another.

Example 1.9. Let $\unicode[STIX]{x1D706}_{0}$ be the standard contact form on the unit sphere $M=\mathbb{S}^{2n-1}\subset \mathbb{R}^{2n}$ . The time- $t$ Reeb flow of $\unicode[STIX]{x1D706}_{0}$ is

$$\begin{eqnarray}(z_{1},\ldots ,z_{n})\mapsto (e^{i2t}z_{1},\ldots ,e^{i2t}z_{n})\end{eqnarray}$$

and so every Reeb orbit is closed with (minimal) period $\unicode[STIX]{x1D70B}$ . For the choice $T=\unicode[STIX]{x1D70B}$ , the set ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},e}(\unicode[STIX]{x1D70B})$ is then a rigid constellation (diffeomorphic to $\mathbb{S}^{2n-1}$ ) with $T^{+}=2\unicode[STIX]{x1D70B}$ .

Example 1.10. Consider an $n$ -tuple $\mathbf{r}=(r_{1},\ldots ,r_{n})$ in $\mathbb{R}^{n}$ such that

$$\begin{eqnarray}0<r_{1}<r_{2}<\cdots <r_{n}<\sqrt{2}r_{1}.\end{eqnarray}$$

Equip the unit sphere $\mathbb{S}^{2n-1}$ with the contact form

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\mathbf{r}}(z)=\biggl(\frac{|z_{1}|^{2}}{r_{1}^{2}}+\cdots +\frac{|z_{n}|^{2}}{r_{n}^{2}}\biggr)^{-1}\unicode[STIX]{x1D706}_{0}(z),\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{0}$ is the form from Example 1.9. The time- $t$ Reeb flow of $\unicode[STIX]{x1D706}_{\mathbf{r}}$ is

$$\begin{eqnarray}(z_{1},\ldots ,z_{n})\mapsto (e^{i2t/r_{1}^{2}}z_{1},\ldots ,e^{i2t/r_{n}^{2}}z_{n}).\end{eqnarray}$$

We then have $T_{\min }=\unicode[STIX]{x1D70B}r_{1}^{2}$ and are free to choose a $T$ of the form $T_{k}=\unicode[STIX]{x1D70B}r_{k}^{2}$ for some $k=1,\ldots ,n.$ For these choices, we have

$$\begin{eqnarray}{\mathcal{C}}_{\unicode[STIX]{x1D706}_{\mathbf{r}},e}(T_{k})=\{\unicode[STIX]{x1D6E4}_{1},\ldots ,\unicode[STIX]{x1D6E4}_{k}\},\end{eqnarray}$$

where each $\unicode[STIX]{x1D6E4}_{i}$ is the $\mathbb{R}/\mathbb{Z}$ -family of closed Reeb orbits with image equal to the intersection of $\mathbb{S}^{2n-1}$ with the $z_{i}$ -plane. Each ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{\mathbf{r}},e}(T_{k})$ is a rigid constellation with

$$\begin{eqnarray}T_{k}^{+}=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D70B}r_{k+1}^{2}\quad & \text{if }k<n,\\ 2\unicode[STIX]{x1D70B}r_{1}^{2}\quad & \text{if }k=n.\end{array}\right.\end{eqnarray}$$

A collection of closed Reeb orbits of a contact form $\unicode[STIX]{x1D706}$ is said to be nondegenerate if each of its elements $\unicode[STIX]{x1D6FE}(t)$ is nondegenerate (and hence isolated) in the usual sense. The rigid constellation in Example 1.9 is degenerate (Morse–Bott nondegenerate) while those described in Examples 1.10 and 1.11 are nondegenerate.

In § 2, we will associate to each nondegenerate rigid constellation ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ a version of $\mathbb{S}^{1}$ -equivariant Floer homology. With this we will define the rank of ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ . At this point we can now state our broadest generalization of Theorem 1.1.

Theorem 1.12. Let $(M,\unicode[STIX]{x1D706}_{0})$ be a closed contact manifold and let ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T)$ be a nondegenerate rigid constellation. Let $\unicode[STIX]{x1D706}=f\unicode[STIX]{x1D706}_{0}$ for some positive function $f$ . If

(4) $$\begin{eqnarray}\frac{\max (f)}{\min (f)}<\min \biggl\{\frac{T^{+}}{T},\frac{T_{\min }(\unicode[STIX]{x1D706}_{0})+T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})}{T}\biggr\}\end{eqnarray}$$

and every closed Reeb orbit of $\unicode[STIX]{x1D706}$ in class $\unicode[STIX]{x1D6FC}$ and with period in the interval

$$\begin{eqnarray}[\min (f)T_{\min },\max (f)T]\end{eqnarray}$$

is nondegenerate, then there are at least $\text{rank}({\mathcal{C}}_{\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC}}(T))$ such orbits which are distinct from one another. These orbits are geometrically distinct from one another if the class $\unicode[STIX]{x1D6FC}$ is either primitive or of infinite order. Otherwise, these orbits are geometrically distinct if there are no (fast) closed Reeb orbits of $\unicode[STIX]{x1D706}$ with period less than or equal to

$$\begin{eqnarray}\frac{1}{|\unicode[STIX]{x1D6FC}|}(T\max (f)-T_{\min }\min (f))\end{eqnarray}$$

that represent a class $\unicode[STIX]{x1D6FD}$ such that $\unicode[STIX]{x1D6FD}^{k}=\unicode[STIX]{x1D6FC}$ for some integer $k>1$ .

The primary point of this result is that it establishes the rigidity of closed Reeb orbits without assumptions on the ambient contact manifold. In particular, it does not assume the existence of strong symplectic fillings. The price of this generality is the presence of the term

$$\begin{eqnarray}\frac{T_{\min }(\unicode[STIX]{x1D706}_{0})+T_{\min }(\unicode[STIX]{x1D706}_{0},\unicode[STIX]{x1D6FC})}{T}.\end{eqnarray}$$

While this term constrains the range of the rigidity phenomenon, it is needed to achieve compactness for the relevant moduli spaces of Floer trajectories used here to detect it. Despite the fact that the term is expressed in the language of dynamics, we do not know whether it represents an actual boundary to the generalized Ekeland–Lasry rigidity phenomenon. When $(M,\unicode[STIX]{x1D706}_{0})$ does admit strong symplectic fillings the proof of Theorem 1.12 simplifies greatly and yields rigidity results with larger, sometimes infinite, range. These results are described below in § 1.5.

In Theorem 1.12 we have imposed nondegeneracy assumptions on certain closed Reeb orbits of $\unicode[STIX]{x1D706}$ while in Theorem 1.6 no such assumptions are made. The lower bounds of Theorem 1.6 correspond to cup-length estimates in the spirit of those predicted in the strong form of Arnold’s conjecture for Hamiltonian diffeomorphisms. Theorem 1.12 is more in the spirit of the nondegenerate form of Arnold’s conjecture. Note that rather than assuming the nondegeneracy of all closed Reeb orbits of $\unicode[STIX]{x1D706}$ (strong nondegeneracy) we have only assumed nondegeneracy for the closed Reeb orbits of $\unicode[STIX]{x1D706}$ in a fixed range of periods. For the Ekeland–Lasry rigidity phenomenon, this seems to be the appropriate assumption to obtain Morse-type inequalities. The difference between this nondegeneracy assumption and the strong form is analogous, in the setting of Arnold’s conjecture, to the difference between the assumption that the fixed points of a Hamiltonian diffeomorphism are nondegenerate and the assumption (clearly irrelevant in that case) that all its periodic points are nondegenerate. This point its captured nicely by the contact forms $\unicode[STIX]{x1D706}_{\mathbf{r}}$ from Example 1.10. Given $\mathbf{r}=(r_{1},r_{2})$ with $r_{1}\leqslant r_{2}<\sqrt{2}r_{1}$ , we know that all contact forms $\unicode[STIX]{x1D706}_{\mathbf{r}}$ have at least two closed Reeb orbits with periods in $[\unicode[STIX]{x1D70B}r_{1}^{2},\unicode[STIX]{x1D70B}r_{2}^{2}]$ and of course we would like our theorems to see this too. For $r_{1}=r_{2}$ , these orbits are detected by Theorem 1.3 and, for $r_{1}<r_{2}$ , they are detected by Theorem 1.12. In contrast, contact forms $\unicode[STIX]{x1D706}_{\mathbf{r}}$ satisfy the strong nondegeneracy assumption if and only if $r_{1}$ and $r_{2}$ are rationally independent.

1.4 Applications (with no strong symplectic fillings)

Theorem 1.12 can be applied to any Reeb flow on a closed manifold for which one has a basic understanding of its fastest closed orbits. One does not need the contact manifold to admit a strong symplectic filling, nor must one preclude the existence of certain closed Reeb orbits. These points are illustrated by the following three applications.

$(\mathbf{A})$ Consider the three-dimensional torus $\mathbb{T}^{3}=\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}$ with angular coordinates $(x,y,\unicode[STIX]{x1D703})$ and the familiar family of contact forms

(5) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{k}=\cos (k\unicode[STIX]{x1D703})\,dx+\sin (k\unicode[STIX]{x1D703})\,dy\end{eqnarray}$$

for $k\in \mathbb{N}$ . The underlying contact structures $\unicode[STIX]{x1D709}_{k}=\ker \unicode[STIX]{x1D706}_{k}$ are all weakly symplectically fillable (and hence tight). However, Eliashberg proved in [Reference EliashbergEli96] that for $k\geqslant 2$ the contact structures $\unicode[STIX]{x1D709}_{k}$ are not strongly symplectically fillable. Hence, one cannot define symplectic cohomology or linearized contact homology for the contact forms they support. Nevertheless, Theorem 1.12 can be applied to all such forms.

The Reeb vector field of $\unicode[STIX]{x1D706}_{k}$ is

$$\begin{eqnarray}\cos (k\unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}+\sin (k\unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}.\end{eqnarray}$$

Hence, the Reeb flow is linear on each $xy$ -torus. The closed Reeb orbits represent nonzero classes of the form

$$\begin{eqnarray}(m,n,0)\in \mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}=\text{H}_{1}(\mathbb{T}^{3};\mathbb{Z}).\end{eqnarray}$$

For a fixed pair of integers $(m,n)$ these orbits all have period $\sqrt{m^{2}+n^{2}}$ . They foliate $k$ subtori of the form

$$\begin{eqnarray}\mathbb{T}^{2}\times \{\unicode[STIX]{x1D703}_{m,n}^{j}\}\subset \mathbb{T}^{3},\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{m,n}^{1},\ldots ,\unicode[STIX]{x1D703}_{m,n}^{k}$ are the solutions of the two equations

$$\begin{eqnarray}\cos (k\unicode[STIX]{x1D703})=\frac{m}{\sqrt{m^{2}+n^{2}}}\end{eqnarray}$$

and

$$\begin{eqnarray}\sin (k\unicode[STIX]{x1D703})=\frac{n}{\sqrt{m^{2}+n^{2}}}.\end{eqnarray}$$

The orbits on these tori are simple if and only if $m$ and $n$ are relatively prime. In this case, for $T=\sqrt{m^{2}+n^{2}}$ the collection ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{k},(m,n,0)}(T)$ is a rigid constellation with $T^{+}=+\infty$ . (Here we have identified $\text{H}_{1}(\mathbb{T}^{3};\mathbb{Z})$ with $[\mathbb{S}^{1},\mathbb{T}^{3}]$ in the obvious way.) A simple perturbation argument, see for example [Reference BourgeoisBou03], shows that each torus contributes $2$ to the rank of ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{k},(m,n,0)}(T)$ . Hence, $\text{rank}({\mathcal{C}}_{\unicode[STIX]{x1D706}_{k},(m,n,0)}(T))=2k$ . With this, Theorem 1.12 implies the following result.

Theorem 1.14. Let $\unicode[STIX]{x1D706}=f\unicode[STIX]{x1D706}_{k}$ , where $f:\mathbb{T}^{3}\rightarrow \mathbb{R}$ is a positive function. Let $m$ and $n$ be relatively prime integers. If

$$\begin{eqnarray}\frac{\max (f)}{\min (f)}<\frac{\sqrt{m^{2}+n^{2}}+1}{\sqrt{m^{2}+n^{2}}}\end{eqnarray}$$

and the closed Reeb orbits of $\unicode[STIX]{x1D706}$ in class $(m,n,0)$ and with period in the interval

$$\begin{eqnarray}[\min (f)\sqrt{m^{2}+n^{2}},\max (f)\sqrt{m^{2}+n^{2}}]\end{eqnarray}$$

are nondegenerate, then there are at least $2k$ such orbits which are geometrically distinct from one another.

$(\mathbf{B})$ Next we apply Theorem 1.12 to an overtwisted contact three-manifold. Consider the manifold $M=\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}\times [0,2\unicode[STIX]{x1D70B}]$ with coordinates $(x,y,t)$ and the family of contact forms

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{k}=\cos (kt)\,dx+\sin (kt)\,dy\end{eqnarray}$$

for $k\in \mathbb{N}\cup \{0\}$ . Each $\unicode[STIX]{x1D702}_{k}$ is invariant under the free $\mathbb{R}/\mathbb{Z}$ -action generated by the vector field $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y$ . Thus, we can perform the contact cut operation, as defined by Lerman in [Reference LermanLer01], with respect to the restriction of this action to the boundary of $M$ (see [Reference LermanLer01, Example 2.12]). We obtain in this way contact forms $\widetilde{\unicode[STIX]{x1D702}}_{k}$ on $\mathbb{R}/\mathbb{Z}\times \mathbb{S}^{2}$ . Here $x$ can still be viewed as a coordinate parameterizing the $\mathbb{R}/\mathbb{Z}$ -fibres of the product $\mathbb{R}/\mathbb{Z}\times \mathbb{S}^{2}$ and we can identify $\widetilde{\unicode[STIX]{x1D702}}_{k}$ with $\unicode[STIX]{x1D702}_{k}$ away from the fibres over the poles of $\mathbb{S}^{2}$ (which correspond to $t=0$ and $t=2\unicode[STIX]{x1D70B}$ ). As shown in [Reference LermanLer01], the contact structures $\ker (\widetilde{\unicode[STIX]{x1D702}}_{k})$ are overtwisted and contactomorphic to one another for all $k>0$ . Hence, the contact manifolds $(\mathbb{R}/\mathbb{Z}\times \mathbb{S}^{2},\widetilde{\unicode[STIX]{x1D702}}_{k})$ have no strong symplectic fillings for $k>0$ .

The Reeb vector field of $\widetilde{\unicode[STIX]{x1D702}}_{k}$ ( $\unicode[STIX]{x1D702}_{k}$ ) is

$$\begin{eqnarray}\cos (kt)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}+\sin (kt)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\end{eqnarray}$$

and so the closed Reeb orbits in class

$$\begin{eqnarray}m\in \mathbb{Z}=\text{H}_{1}(\mathbb{R}/\mathbb{Z}\times \mathbb{S}^{2};\mathbb{Z})\end{eqnarray}$$

occur for values of $t$ such that

(6) $$\begin{eqnarray}\cos (kt)=\frac{m}{\sqrt{m^{2}+n^{2}}}\end{eqnarray}$$

and

(7) $$\begin{eqnarray}\sin (kt)=\frac{n}{\sqrt{m^{2}+n^{2}}}\end{eqnarray}$$

for some $n\in \mathbb{Z}.$ For a solution $t$ of these equations located in $(0,2\unicode[STIX]{x1D70B})$ , the corresponding orbits occur in an $\mathbb{S}^{1}$ -family that foliates the two-dimensional $xy$ -tori, $\mathbb{T}^{2}\times \{t\}$ , as in the previous example. For $t=0$ and $t=2\unicode[STIX]{x1D70B}$ , the corresponding orbits are isolated and correspond to the fibres of $\mathbb{R}/\mathbb{Z}\times \mathbb{S}^{2}$ over the poles of $\mathbb{S}^{2}$ .

The fastest simple closed orbits of $\widetilde{\unicode[STIX]{x1D702}}_{k}$ all have period $2\unicode[STIX]{x1D70B}$ and correspond to the four cases $(m,n)=(\pm 1,0)$ and $(m,n)=(0,\pm 1)$ . For the case $(m,n)=(1,0)$ , there are $k+1$ solutions to (6) and (7),

$$\begin{eqnarray}0,\frac{2\unicode[STIX]{x1D70B}}{k},\ldots ,\frac{2\unicode[STIX]{x1D70B}(k-1)}{k},2\unicode[STIX]{x1D70B}.\end{eqnarray}$$

As described above, the values $t=0$ and $t=1$ correspond to isolated simple closed Reeb orbits, and the other $k-1$ values of $t$ correspond to subtori foliated by simple closed Reeb orbits (in class $(1,0)$ ). The collection of all these orbits forms a rigid constellation of rank $2k$ with $T=2\unicode[STIX]{x1D70B}$ and $T^{+}=2\sqrt{2}\unicode[STIX]{x1D70B}$ .

For the case $(m,n)=(-1,0)$ , there are $k$ solutions of (6) and (7),

$$\begin{eqnarray}\frac{\unicode[STIX]{x1D70B}}{k},\frac{3\unicode[STIX]{x1D70B}}{k},\ldots ,2\unicode[STIX]{x1D70B}-\frac{\unicode[STIX]{x1D70B}}{k}.\end{eqnarray}$$

Each of these corresponds to unique subtori foliated by simple closed Reeb orbits (in class $(-1,0)$ ). The collection of these tori again forms a rigid constellation of rank $2k$ with $T=2\unicode[STIX]{x1D70B}$ and $T^{+}=2\sqrt{2}\unicode[STIX]{x1D70B}$ .

Finally, for the cases $(m,n)=(0,\pm 1)$ , there are again $k$ solutions of (6) and (7) each of which corresponds to a subtorus foliated by simple closed Reeb orbits which are contractible in $\mathbb{R}/\mathbb{Z}\times \mathbb{S}^{2}$ . These also form a rigid constellation of rank $2k$ , this time with $T=2\unicode[STIX]{x1D70B}$ and $T^{+}=4\unicode[STIX]{x1D70B}$ .

Applying Theorem 1.12 to these three rigid constellations, we get the following rigidity theorem.

$(\mathbf{C})$ Now we apply Theorem 1.12 to a family of overtwisted contact forms on $\mathbb{S}^{3}$ . We start again with the manifold $M=\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}\times [0,2\unicode[STIX]{x1D70B}]$ with coordinates $(x,y,t)$ . For $k\in \mathbb{N}\cup \{0\}$ , we now consider the contact forms

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{k}=\cos \biggl(\biggl(k+\frac{1}{4}\biggr)t\biggr)\,dx+\sin \biggl(\biggl(k+\frac{1}{4}\biggr)t\biggr)\,dy.\end{eqnarray}$$

Each $\unicode[STIX]{x1D701}_{k}$ is invariant under the free $\mathbb{R}/\mathbb{Z}$ -actions generated by the vector fields $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x$ and $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y$ . This time we perform a different contact cut operation by choosing the $\mathbb{R}/\mathbb{Z}$ -action generated by $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y$ at the boundary component $\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}\times \{0\}$ , and the $\mathbb{R}/\mathbb{Z}$ -action generated by $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x$ at the boundary component $\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}\times \{2\unicode[STIX]{x1D70B}\}$ . The resulting manifold is $\mathbb{S}^{3}$ and we denote the resulting contact form by $\widetilde{\unicode[STIX]{x1D701}}_{k}$ . For all $k>0$ , the contact structures $\ker (\widetilde{\unicode[STIX]{x1D701}}_{k})$ are overtwisted and contactomorphic to one another (see the proof of [Reference LermanLer01, Theorem 3.1]).

Away from the fibres of the Hopf fibration that lie over the two poles in $\mathbb{S}^{2}$ , we can identify $\widetilde{\unicode[STIX]{x1D701}}_{k}$ with $\unicode[STIX]{x1D701}_{k}$ . Thus, we can analyze the closed Reeb orbits as before. Here the set of the fastest simple closed orbits of $\widetilde{\unicode[STIX]{x1D701}}_{k}$ is a rigid constellation with $T=2\unicode[STIX]{x1D70B}$ , $T^{+}=2\sqrt{2}\unicode[STIX]{x1D70B}$ . It comprises two isolated orbits and $4k$ $\mathbb{S}^{1}$ -families of closed orbits and so has rank $8k+1$ . In this setting Theorem 1.12 implies the following result.

1.5 Applications (with strong symplectic fillings)

For contact manifolds which admit a symplectic filling, Theorems 1.6 and 1.12 both hold as stated but with improved ranges (if $\unicode[STIX]{x1D6FC}$ is interpreted as a free homotopy class of the filling). For example, if we assume that the prequantization space $(M,\unicode[STIX]{x1D706}_{Q})$ admits an exact symplectic filling, then Theorem 1.6 holds, as stated, with the larger upper bound

$$\begin{eqnarray}\frac{\max (f)}{\min (f)}<|\unicode[STIX]{x1D6FC}_{\mathbf{f}}|+1.\end{eqnarray}$$

Similarly, assuming that there is an exact symplectic filling of $(M,\unicode[STIX]{x1D706}_{0})$ , Theorem 1.12 holds, as stated, with the upper bound

$$\begin{eqnarray}\frac{\max (f)}{\min (f)}<\frac{T^{+}}{T}.\end{eqnarray}$$

As the next set of applications demonstrates, this version of Theorem 1.12 allows one to view a variety of previously known rigidity phenomena from a single perspective.

$(\mathbf{D})$ Applying Theorem 1.12 to the contact forms in Example 1.10, one can recover the various pinching theorems of Berestycki et al. established in [Reference Berestycki, Lasry, Mancini and RufBLMR85] up to some extra nondegeneracy assumptions.

$(\mathbf{E})$ Returning to the setting of Example 1.11, consider a closed Riemannian manifold $(B,g)$ with negative sectional curvature and the corresponding unit cosphere bundle $\unicode[STIX]{x1D6F4}_{g^{\ast }}\subset T^{\ast }B$ . Let $\tilde{\unicode[STIX]{x1D6FC}}$ be a nontrivial primitive element of $[\mathbb{R}/\mathbb{Z},B]$ and denote its lift to $[\mathbb{R}/\mathbb{Z},\unicode[STIX]{x1D6F4}_{g^{\ast }}]$ by $\unicode[STIX]{x1D6FC}$ . As described in Example 1.11, the rigid constellation ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{g^{\ast }},\unicode[STIX]{x1D6FC}}(T_{\min }(\unicode[STIX]{x1D706}_{g^{\ast }},\unicode[STIX]{x1D6FC}))$ has $T^{+}=+\infty$ . It also has rank $1$ and so Theorem 1.12 implies that every hypersurface $\unicode[STIX]{x1D6F4}$ in $T^{\ast }B$ which is fibrewise star-shaped about the zero section must carry at least one closed characteristic in class $\unicode[STIX]{x1D6FC}$ . A similar but stronger result in this direction is proved in [Reference Biran, Polterovich and SalamonBPS03] as Corollary 3.4.2 (see also [Reference Hofer and ViterboHV88]).

$(\mathbf{F})$ Let $g_{n}$ be the standard flat metric on the $n$ -dimensional torus $\mathbb{T}^{n}$ . Denote its cosphere bundle by $\unicode[STIX]{x1D6F4}_{g_{n}^{\ast }}\subset T^{\ast }\mathbb{T}^{n}$ and the restriction of the tautological one-form to $\unicode[STIX]{x1D6F4}_{g_{n}^{\ast }}$ by $\unicode[STIX]{x1D706}_{g_{n}^{\ast }}$ . Every nontrivial primitive class $\tilde{\unicode[STIX]{x1D6FC}}$ in $[\mathbb{R}/\mathbb{Z},\mathbb{T}^{n}]$ lifts to a class $\unicode[STIX]{x1D6FC}$ in $[\mathbb{R}/\mathbb{Z},\unicode[STIX]{x1D6F4}_{g_{n}^{\ast }}]$ , which is again nontrivial and primitive. There is a unique $\mathbb{T}^{n}$ -family of closed Reeb orbits in class $\unicode[STIX]{x1D6FC}$ . Denoting their period by $T(\unicode[STIX]{x1D6FC})$ , the constellation ${\mathcal{C}}_{\unicode[STIX]{x1D706}_{g_{n}^{\ast }},\unicode[STIX]{x1D6FC}}(T(\unicode[STIX]{x1D6FC}))$ is rigid with $T^{+}=+\infty$ and $\text{rank}({\mathcal{C}}_{\unicode[STIX]{x1D706}_{g_{n}^{\ast }},\unicode[STIX]{x1D6FC}}(T))=2^{n}$ . With this, Theorem 1.12 implies the following.

This complements a theorem of Cielebak from [Reference CieliebakCie94], which establishes the existence of at least

$$\begin{eqnarray}\biggl\lfloor\frac{n}{2}\biggr\rfloor+1\end{eqnarray}$$

geometrically distinct closed Reeb orbits with no nondegeneracy assumption.

$(\mathbf{G})$ The finite collection of prime closed geodesics in Katok’s famous examples of Finsler metrics from [Reference KatokKat73] also yields a rich source of rigid constellations. Consider for example a Katok metric $g_{K}$ on the sphere $P=\mathbb{S}^{2n}$ or $\mathbb{S}^{2n-1}$ . It has precisely $2n$ prime closed geodesics and can be constructed so that for any $\unicode[STIX]{x1D716}\in (0,1)$ the longest prime geodesic has length $1+\unicode[STIX]{x1D716}$ and the shortest has length $1-\unicode[STIX]{x1D716}$ . For $\unicode[STIX]{x1D716}<\frac{1}{2}$ and $T=1+\unicode[STIX]{x1D716}$ , the collection of prime closed geodesics is then a rigid constellation with $T^{+}=2(1-\unicode[STIX]{x1D716})$ and rank $2n$ . If, as above, we let $\unicode[STIX]{x1D6F4}_{g_{K}^{\ast }}\subset T^{\ast }P$ be the cosphere bundle of $g_{K}$ , then for any other hypersurface $\unicode[STIX]{x1D6F4}$ of $T^{\ast }\mathbb{S}^{2n}$ which is fibrewise star-shaped about the zero section there is a unique smooth function $f_{\unicode[STIX]{x1D6F4}}:\unicode[STIX]{x1D6F4}_{g_{K}^{\ast }}\rightarrow (0,\infty )$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}=\{(q,f_{\unicode[STIX]{x1D6F4}}(q,p)p):(q,p)\in \unicode[STIX]{x1D6F4}_{g_{K}^{\ast }}\}.\end{eqnarray}$$

Let $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6F4}}$ be the restriction of the tautological one-form to $\unicode[STIX]{x1D6F4}$ . In this setting Theorem 1.12 implies the following result.

Theorem 1.18. If

$$\begin{eqnarray}\frac{\max (f_{\unicode[STIX]{x1D6F4}})}{\min (f_{\unicode[STIX]{x1D6F4}})}<\frac{2(1-\unicode[STIX]{x1D716})}{1+\unicode[STIX]{x1D716}}\end{eqnarray}$$

and all the closed Reeb orbits of $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6F4}}$ with period in the interval

$$\begin{eqnarray}\biggl[\frac{2\unicode[STIX]{x1D70B}\min (f_{\unicode[STIX]{x1D6F4}})}{1+\unicode[STIX]{x1D716}},\frac{2\unicode[STIX]{x1D70B}\max (f_{\unicode[STIX]{x1D6F4}})}{1-\unicode[STIX]{x1D716}}\biggr]\end{eqnarray}$$

are nondegenerate, then $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6F4}}$ has at least $2n$ distinct closed Reeb orbits with periods in this interval. These orbits are geometrically distinct if there are no closed Reeb orbits of $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6F4}}$ with period less than or equal to the length of this interval.

If one restricts Theorem 1.18 to hypersurfaces that are cosphere bundles of Finsler metrics, it then yields multiplicity results for closed geodesics which complement recent work of Rademacher in [Reference RademacherRad07] and of Wang in [Reference WangWan15].

1.6 Fast orbits

The possible presence of periodic orbits with small periods sometimes obstructs our ability to conclude that the distinct orbits detected by Theorems 1.6 and 1.12 are in fact geometrically distinct. This is a fundamental, and somewhat notorious, difficulty common to such multiplicity problems. In certain settings one can preclude the existence of such fast orbits by imposing geometric restrictions, like the convexity of the hypersurfaces in [Reference Ekeland and LasryEL80]. It is natural then to ask if one can find conditions on the contact structure which preclude fast orbits and are invariant under contactomorphisms. We prove here that this is not possible. This is one implication of our final theorem, which can also be viewed as a soft compliment to Theorems 1.6 and 1.12.

There are some intriguing phenomena hidden in the gap between the construction underlying Theorem 1.19 and, say, Theorem 1.12. For example, in creating the nearby form $\unicode[STIX]{x1D706}$ in Theorem 1.19 with an arbitrarily fast closed orbit we are forced, by the Reeb condition, to cede all control over the number and basic properties of any additional closed Reeb orbits we might also create in the process. This follows from work of Rechtman in [Reference RechtmanRec09] and is discussed further in Remark 5.5.

1.7 Related works

Our proof of Theorem 1.6 is motivated by the method to obtain cup-length estimates in Floer theory that was introduced by Albers and Momin in [Reference Albers and MominAM10] and further developed by Albers and Hein in [Reference Albers and HeinAH16].

The proof of Theorem 1.12 has two main parts. The first involves the construction of a version of Hamiltonian Floer theory for rigid constellations. In this construction we use several results concerning the Hamiltonian Floer theory of autonomous Hamiltonians established by Bourgeois and Oancea in [Reference Bourgeois and OanceaBO09]. To obtain the $C^{0}$ -bounds for our Floer trajectories (which allows us to do away with fillings), we also adapt an argument of Albers et al. from [Reference Albers, Fuchs and MerryAFM15]. In the second part of the proof we use the Floer-theoretic tools developed in the first to adapt an argument of Chekanov from [Reference ChekanovChe98] to detect the desired closed orbits. This is based on previous work of the author from [Reference KermanKer08].

Similar ideas to those underlying Theorem 1.12 were developed by Gutt in his thesis [Reference GuttGut14] and subsequent paper [Reference GuttGut15].Footnote ¹ In these works, Gutt shows that positive $\mathbb{S}^{1}$ -equivariant symplectic homology can be used as a contact invariant for a certain class of fillable contact manifolds that can be realized as the boundary of Liouville domains. Among the many interesting applications of his theory, Gutt reproves Theorem 1.1 under the additional strong nondegeneracy assumption, and also proves a result [Reference GuttGut15, Theorem 1.6] very similar in content to Theorem 1.6 here. Happily, besides a shared debt owed to the technical foundations for $\mathbb{S}^{1}$ -equivariant Hamiltonian Floer theory laid down by Bourgeois and Oancea, this is essentially the extent of the overlap between the two projects.

The construction of a Reeb semi-plug described in the proof of Theorem 1.19 is similar in several details to Cieliebak’s construction of a confoliation-type plug from [Reference CieliebakCie97]. The goals of the two constructions diverge at an early stage, however. In keeping within the class of Reeb flows we sacrifice here the possibility, achieved in [Reference CieliebakCie97], of realizing the insertion of our plugs on fillable contact manifolds, as the result of a symplectomorphism acting on the interior of a filling.