Pedro Salamão

(joint work with Umberto Hryniewicz and Joan Licata)

Question: determine the topology of a co-oriented contact manifold from the Reeb flow of defining contact form.

Motivating example: characterisation of the tight 3-sphere. If $\alpha$ is dynamically convex & non-degenerate and admits a closed Reeb orbit $P$ so that $P$ is unknotted, has self-linking number $-1$ and the CZ index is $3$. Then $M$ is the tight 3-sphere. (Converse also true.)

(If, $P$ as above, then is binding of an open book decomposition of $M$ with disk-like pages. This theorem is proved by constructing this open book.)

$L(p,q)$ with standard contact structure $\xi_{\mathrm{std}}$.

Let $K$ be a knot. Let $p \ge 1$. Then a $p$-disk for $K$ is a disk whose interior embeds in $M \setminus K$ and $u|_{\partial D}$ is a $p$-cover of $K$.

If $K$ admits a $p$-disk, it is $p$-unknotted and is an order $p$ rational unknot.

The image of the Hopf fibres project to $p$-unknots $K_0$ and $K_1$ inside $L(p,q)$ with self-linking number $-p$.

If you look at the $p$-disk with boundary on $K_0$, and go around the boundary annulus once, you come back to a sheet that is shifted over by $-q$. (Call this the monodromy.)

For $K_1$ the monodromy is $q'$ where $qq' = 1 (mod p)$.

Theorem:  (Hryniewicz, Licata, Salamão)
 $(M, \xi)$ with $c_1(\xi) = 0$ on $\pi_2(M)$. Let $p \in \mathbb N^*$

Then, contactomorphic to $L(p,q), \xi_{\mathrm{std}}$ if and only if $\xi$ is the kernel of a non-degen contact form $\alpha$ admitting a prime closed Reeb orbit $K$ such that

• $K$ is p-unknotted, $CZ( K^p ) \ge 3$, self-linking $K^p = -p$ and monodromy is $-q$
• every contractible Reeb orbit $P \subset M \setminus K$ with CZ index 2 is not contractible in $M \setminus K$.

Strategy of proof: make $p$ disk for $K$ become a fast pseudoholomorphic plane with $\pi du \ne 0$. Then, foliate complement with these planes. Technical condition on index $2$ orbits gives rational open book and then result.

Great idea: $C^0$ perturb the disk we start with, to obtain the following picture:

• characteristic foliation has only one nicely elliptic singular point $e$ and all leaves reach the boundary transversely (recall the boundary is a transverse knot)
• transverse to $R$ near the boundary (this uses the fact the CZ index is at least 3 for the $p$-fold cover of the knot)
• the disk does not contain any periodic orbits of the Reeb flow (i.e. only one is the boundary)
• Local model near $e$ with nice Darboux coordinates.

Ah, idea now is to consider the Bishop disks near singularity. One boundary component is the constant disk in $e$. Other one is a building. Chasing through what happens, must break on orbits with index at least $2$. This index $2$ orbit condition allows us to exclude most of what can happen. From this, we conclude that the other boundary component of the building is the plane we want.

Additional applications:

Geodesic flows and restricted planar 3 body problem: both live on $\mathbb RP^3 = L(2,1)$.

Surface of revolution: double cover of meridians have CZ index 3 or 4. Linking argument means hypotheses satisfied for other orbits.

Finsler metrics with pinched flag curvature (pinching based on reversibility). Paternain says is dynamically convex and Rademacher says there is a simple closed geodesic. Some work says this satisfies hypotheses of the theorem.

planar restricted 3 body problem with circular earth-moon orbit.
Albers, Fish, Frauenfelder, Hofer, Paternain, von Koert: subsets of these in various papers show that low energy gives two components each of which regularizes to a Reeb flow on $L(2,1)$. Obtain disk like surfaces of section for component near the moon (by working on $S^3$). Also show dynamical convexity but not geometric convexity near the first critical value for rotating Kepler problem. Retrograde orbit has index $3$ but the direct orbit has higher index.

Possible application of this rational open book stuff: work with prescribed orbit with high index, or for perturbations that destroy dynamical convexity.