## A guest post by Dans C-G and P

[The following is a guest post by Dan Cristofaro-Gardiner and Dan Pomerleano. If anyone else is interested in contributing a guest post, please feel free to contact me. A blog is a good outlet for short or informal mathematical thoughts which might not have a place in a traditional publication, and guest posting is convenient if you are not yet ready to start your own blog. -M.H.]

What can we say about the minimum number of Reeb orbits?

The paper From one Reeb orbit to two showed that any Reeb flow on a closed contact three-manifold must have at least two closed orbits. While examples exist with exactly two orbits (e.g. irrational ellipsoids), there is no known example of a contact manifold that is not a lens space where the Reeb flow has finitely many closed orbits. It is therefore natural to try to refine this result under additional assumptions, and there has been interesting work in this direction by Hofer-Wysocki-Zehnder, Colin-Honda, Ginzburg-Gurel-Macarini, and others.

One example of such a refinement is a theorem of Hutchings and Taubes, which states that, for a nondegenerate contact form, the Reeb flow must have at least three distinct embedded Reeb orbits on any manifold that is not a lens space. It turns out that if the contact structure is not torsion, one can slightly improve on this result:

Proposition 1. Let $(Y,\lambda)$ be a closed contact three-manifold, and let $\xi$ be the contact structure for $\lambda$. Assume that $c_1(\xi)$ is not torsion. Then the Reeb flow has at least three distinct embedded orbits. If $\lambda$ is nondegenerate, then the Reeb flow has at least four distinct embedded orbits.

The proof of this proposition is given below. The arguments are similar to those in “From one Reeb orbit to two”, so this post may also be of interest to anyone curious about that paper.

1. Spectral invariants and a review of ECH

Our proof (as well as the proof in “From one Reeb orbit to two”) uses the “spectral invariants” defined by Hutchings in Quantitative embedded contact homology. To recall their definition, let us begin by stating some basic facts about ECH under the assumption that $\lambda$ is nondegenerate. Fix a class $\Gamma \in H_1(Y)$. The group $ECH(Y,\lambda,\Gamma)$ is the homology of a chain complex $ECC$. This chain complex is generated by orbit sets $\alpha = \lbrace (\alpha_i,m_i) \rbrace$, where the $\alpha_i$ are distinct embedded Reeb orbits, the $m_i$ are positive integers, and the total homology class of $\alpha$ is equal to $\Gamma$. The orbit sets are required to be admissible, which means that each $m_i$ is equal to $1$ when $\alpha_i$ is hyperbolic. It is known that ECH is an invariant of the contact structure $\xi$ (in fact, it is known that ECH is an invariant of the three-manifold, but we will not need this). Thus, the group $ECH(Y,\xi,\Gamma)$ is well-defined.

Let $\sigma$ be a nonzero class in $ECH(Y,\xi,\Gamma)$. We can define invariants $c_{\sigma}(\lambda)$ for any contact form $\lambda$ in the contact structure $\xi$. This works as follows. An orbit set has a symplectic action defined by $\mathcal{A}(\lbrace (\alpha_i,m_i) \rbrace) = \sum_i m_i \int_{\alpha_i} \lambda$. If $\lambda$ is nondegenerate, define $c_{\sigma}(\lambda)$ to be the “minimum symplectic action” required to represent the class $\sigma$. If $\lambda$ is degenerate, define $c_{\sigma}(\lambda) = \lim_{n \to \infty} c_{\sigma}(\lambda_n)$, where $\lambda_n$ are a sequence of nondegenerate contact forms converging in $C^0$ to $\lambda$. This works essentially because the $c_{\sigma}(\cdot)$ behave like symplectic capacities: they satisfy monotonicity and scaling axioms which make $c_{\sigma}(\lambda)$ in the degenerate case well-defined. For the details, see for example “Quantitative embedded contact homology”.

Here is the key fact that we need about spectral invariants:

Fact 2. Let $(Y,\lambda)$ be a (possibly degenerate) contact manifold. Let $\sigma \in ECH(Y,\xi,\Gamma)$. Then $c_{\sigma}(\lambda)=\mathcal{A}(\alpha)$, where $\alpha$ is some orbit set for $\lambda$ with total homology class $\Gamma$. If $\lambda$ is nondegenerate, then $\alpha$ is admissible.

This is proved similarly to Lemma 3.1(a) in “From one Reeb orbit to two”. The proof in the degenerate case uses a standard compactness argument for Reeb orbits of bounded action.

The idea of the proof of the proposition is now to look at the spectral invariants associated to a certain sequence of classes with gradings tending to infinity. If there are too few Reeb orbits, we will find a contradiction with known facts about the asymptotics of these spectral invariants.

2. U-sequences

To make this precise, we now introduce the notion of a “U-sequence”. Recall that ECH comes equipped with a “U-map”, which is a degree $-2$ map defined by counting $I=2$ curves. Also recall that Taubes showed that there is a canonical isomorphism

$ECH_*(Y,\lambda,\Gamma) \cong \widehat{HM}^{-*}(Y,s_{\xi} + PD(\Gamma))$,

where $\widehat{HM}$ denotes the Seiberg-Witten Floer cohomology defined by Kronheimer and Mrowka. The $U$-map agrees with an analogous structure on $\widehat{HM}$ under this isomorphism.

Let $\Gamma$ be a class in $H_1(Y)$. If $c_1(\xi) + 2PD(\Gamma)$ is torsion, then $ECH(Y,\xi,\Gamma)$ has a relative $\mathbb{Z}$ grading. It follows from the above isomorphism together with known facts about $\widehat{HM}$ that this group is infinitely generated. In fact, it is well-known (by again using this isomorphism) that one can always find a U-sequence, namely a sequence of non-zero classes $\sigma_k \in ECH(Y,\xi,\Gamma)$ with definite gradings such that $U(\sigma_k) = \sigma_{k-1}$. We will use a refined version of this statement, involving the canonical mod 2 grading on ECH (in this grading, the grading of an orbit set $\alpha$ is $(-1)^{h(\alpha)}$, where $h$ is the number of positive hyperbolic orbits in the orbit set).

Fact 3. Let $(Y,\lambda)$ be a contact manifold. Assume that $c_1(\xi) + 2PD(\Gamma)$ is torsion. Then either:

• we have $b_1(Y)=0$, in which case there is a U-sequence in even grading, or
• $b_1(Y)>0$, in which case there exist U-sequences in both even and odd grading.

This result can be deduced from the discussion in Section 35.1 of Kronheimer and Mrowka’s book “Monopoles and three-manifolds”.

3. A digression about odd contact manifolds

Fact 3 will be used in our proof of Proposition 1, but it has other interesting consequences as well. For example, let us say that a contact three-manifold $(Y,\lambda)$ is “odd” if all closed embedded Reeb orbits are either elliptic or negative hyperbolic. It was asked previously on this blog whether all odd contact manifolds are lens spaces. Corollary 4 below provides some evidence in favour of this. If $(Y,\lambda)$ is odd, then $ECH(Y,\lambda,\Gamma)$ must be concentrated in even degree. We obtain as a corollary of Fact 3 that:

Corollary 4. If $(Y,\lambda)$ is an odd contact manifold, then $b_1(Y)=0$.

4. The proof

Returning to the proof of Proposition 1, we will also need the following facts about the spectral invariants of a U-sequence associated to any contact form $\latex lambda$:

Fact 5.

• Let $\sigma$ be a nonzero class on $ECH$ with $U\sigma \ne 0$. Then $c_{U(\sigma)}(\lambda) < c_{\sigma}(\lambda)$.
• Let $\lbrace \sigma_k \rbrace$ be a U-sequence. Then

$\lim_{k \to \infty} \frac{c_{\sigma_k}(\lambda)^2}{k} = 2vol(Y,\lambda).$

The first item follows from Stokes’ Theorem in the nondegenerate case; when $\lambda$ is degenerate, the key result is a compactness result for pseudoholomorphic currents due to Taubes, see “From one Reeb orbit to two”. The second item follows from the “volume conjecture” proved in “The asymptotics of embedded contact homology capacities”.

We have now laid out all of the necessary machinery to give our proof.

Proof.

The nondegenerate case. Suppose we have exactly three embedded orbits.

Our manifold $(Y,\lambda)$ must not be odd in view of Corollary 4. We will next show that we must have exactly two elliptic orbits. Choose $\Gamma$ such that $c_1(\xi) + 2PD(\Gamma)$ is torsion. If we had zero elliptic orbits, it follows from the definition of the ECH chain complex that $ECH(Y,\lambda,\Gamma)$ would be finitely generated, contradicting (for example) Fact 3. Let $\lbrace \sigma_k \rbrace$ be a U-sequence. If we had one elliptic orbit $e_1$ and two hyperbolic orbits $h_1,h_2$, we would contradict Fact 5. More precisely, the first bullet of Fact 5 together with Fact 2 would imply that $\frac{c_{\sigma_k}(\lambda)^2}{k}$ would have to grow at least linearly with $k$, while the second bullet implies that this cannot occur.

Thus, we can assume that we have two elliptic orbits $e_1,e_2$ and a positive hyperbolic $h$.

[There can’t be three elliptic orbits because this would contradict Theorem 1.2 in The Weinstein conjecture for stable Hamiltonian structures. -Ed.] The key fact is now that since $c_1(\xi)$ is not torsion, $\Gamma$ is also not torsion. The significance of this is as follows. We have an induced map $\mathbb{Z}^2= \mathbb{Z}[e_1] \oplus \mathbb{Z}[e_2] \to H_1(Y)$, which sends $[e_i]$ to the class represented by the Reeb orbit $e_i$. If the kernel has rank zero, then again, $ECH(Y,\lambda,\Gamma)$ would be finitely generated. If the kernel has rank two, then these orbits would represent torsion classes in homology. On the other hand, by Fact 3, we must have a U-sequence in $ECH(Y,\lambda,\Gamma)$ in even degree. This must take the form $e_1^{m_k}e_2^{n_k}$, contradicting our assumption that $c_1(\xi)$ is non-torsion.

It remains to handle the case when the kernel has rank one. In this case, assume that the kernel is generated by some integer vector $(c,d)$, say with $d>0$. Then each generator of our U-sequence $e_1^{m_k}e_2^{n_k}$ must have the form $e_1^{m_0+x_kc}e_2^{n_0+x_kd}$. Because there are infinitely many distinct $e_1^{m_k}e_2^{n_k}$, we must have $c\ge 0$ (otherwise we would have $-n_0\le x_k \le m_0$ for all $k$). Since $c$ and $d$ are nonnegative, the asymptotics of this sequence would again violate the second bullet of Fact 5, since the action of each term in this sequence would have to be bigger than the action of the previous term by at least the minimum of the actions of $e_1$ and $e_2$.

The degenerate case. By “From one Reeb orbit to two”, we have at least two distinct embedded Reeb orbits. So assume that we have exactly two, $\gamma_1$ and $\gamma_2$. We now argue similarly to before. Namely, again consider the U-sequence latex $\lbrace \sigma_k \rbrace$, as well as the induced map $\mathbb{Z}^2= \mathbb{Z}[\gamma_1] \oplus \mathbb{Z}[\gamma_2] \to H_1(Y)$. By Fact 2, this kernel cannot have rank two. By Fact 2, and the first bullet point of Fact 5, the kernel does not have rank $0$. By repeating the argument in the previous paragraph, it also cannot have rank $1$.

QED

[Any ideas for improving the above bounds further? As suggested at the beginning, one might conjecture that if $(Y,\lambda)$ is a closed contact three-manifold, and if $Y$ is not (a sphere or) a lens space, then there are infinitely many Reeb orbits. -Ed.]