## Mean action and the Calabi invariant

I recently posted a new paper, “Mean action and the Calabi invariant“. There is a bit of a story about where this paper comes from; I didn’t try to explain this in the paper, in order to keep things focused, so let me tell the story here.

In some previous posts on this blog, for example here, I discussed the question of short Reeb orbits. Namely, given a contact form $\lambda$ on a closed three-manifold $Y$, what can you say about the minimum action of a Reeb orbit? I conjectured (maybe I should have made this a “question” instead of a “conjecture”) that there always exists a Reeb orbit with action less than or equal to the square root of the volume of $(Y,\lambda)$. That is, the conjecture is that there exists a Reeb orbit $\gamma$ such that

$\int_\gamma\lambda\le\sqrt{\int_Y\lambda\wedge d\lambda}$.

It turns out that this is false, as shown by a counterexample in this paper by Abbondandolo, Bramham, Hryniewicz, and Salamao. It might still be true under additional assumptions, for example for dynamically convex contact forms on $S^3$. Still an interesting question to think about.

Anyway, I previously spent some time trying to decide whether the conjecture is true for a dynamically convex contact form $\lambda$ on $S^3$ as follows. (Vinicius Ramos also contributed to this.) We know by a theorem of Hofer-Wysocki-Zehnder that $\lambda$ is compatible with an open book decomposition in which the pages are disks. The return map of the Reeb flow defines an area-preserving map $\phi$ from the disk to itself. One can then translate the statement about short Reeb orbits of $\lambda$ to a statement about the dynamics of $\phi$. In the case when the binding orbit is elliptic, the statement looks as follows. (It is another interesting question whether you can always arrange for the binding orbit to be elliptic.)

Let $\omega$ denote the standard area form on the disk $D^2$, renormalized to have area $1$. Let $\phi$ be an area-preseriving diffeomorphism of the disk. Assume that on the boundary, $\phi$ is rotation by angle $2\pi\theta_0$, where $\theta_0$ is a real number. We now define an “action function” $f:D^2\to{\mathbb R}$ as follows. Let $\beta$ be a primitive of $\omega$ on $D^2$, and assume that on the boundary, $\beta(\partial_\theta) = 1/(2\pi)$. The function $f$ is now uniquely defined by the conditions $df=\phi^*\beta-\beta$ and $f|_{\partial D^2}=\theta_0$.

In the above open book situation, these data arise as follows: $\theta_0$ is the inverse of the “rotation number” of the linearized Reeb flow around the binding orbit; $\beta$ is the restriction of the contact form $\lambda$ to a page (we have normalized $\lambda$ so that the symplectic area of the page equals $1$); and $f(x)$ is the time that it takes for the Reeb flow starting at the point $x$ on the page to return to the page. In particular, in this case $\theta_0$ and $f$ are positive.

Now Reeb orbits of $\lambda$ correspond to periodic orbits of $\phi$, along with the binding. The binding has symplectic action $1$ (the symplectic area of the page). If $\gamma=(x_1,\ldots,x_d)$ is a periodic orbit of $\phi$, then the symplectic action of the corresponding Reeb orbit is just

${\mathcal A}(\gamma) = \sum_{i=1}^df(x_i)$.

Furthermore, one can compute the contact volume in terms of $f$; one obtains

${vol}(Y,\lambda) = \int_{D^2}f\omega.$

It turns out that the right hand side is a previously studied quantity, called the Calabi invariant of $(\phi,\theta_0)$. In particular it does not depend on the choice of $\beta$, and it defines a homomorphism from the group of pairs $(\phi,\theta_0)$ to the real numbers.

Anyway, the short Reeb orbit conjecture in this case reduces to the following statement: If the Calabi invariant is less than $1$ (so that the binding does not already fulfill the conjecture), then there exists a periodic orbit of $\phi$ such that the square of its action is less than the Calabi invariant. This turns out to be false for some $\lambda$, as shown by the counterexample in [ABHS]. It might still be true in the dynamically convex case; it is not clear (to me at least) exactly what additional conditions on $\phi$ comes from the dynamical convexity of $\lambda$.

My paper proves a weaker statement. Instead of giving an upper bound on the (squared) action of a periodic orbit of $\phi$, it gives an upper bound on a smaller quantity, namely the “mean action”, defined to be the action divided by the period. (Umberto Hryniewicz suggested to me that something like this should be possible.) If $\gamma$ is a periodic orbit of $\phi$, denote its action by ${\mathcal A}(\phi)$ and its period by $d(\phi)$, so that the mean action is ${\mathcal A}(\phi)/d(\phi)$. The main theorem in my paper then says that if the Calabi invariant is less than the boundary rotation number $\theta_0$, then the infimum of the mean action is less than or equal to the Calabi invariant. (I also assumed that the map $\phi$ is a rotation in a neighborhood of the boundary, not just on the boundary; this is probably not really necessary.) The proof uses embedded contact homology, in particular a new filtration on ECH determined a transverse knot, which could be interesting to study more.

In summary, I originally wanted to prove something about the geometry of contact forms on $S^3$ by reducing it to a statement about area-preserving maps of the disk. This failed, in part because what I was trying to prove was false, and in part because area-preserving maps of the disk are in fact quite subtle. So instead, the paper goes in the reverse direction and uses geometry of contact forms on $S^3$ to prove something about area-preserving maps of the disk.

By the way, the paper doesn’t say this (maybe I’ll add a remark in the next version), but I think that one can also use a similar argument to show that the supremum of the mean action is greater than or equal to the Calabi invariant. However this seems less interesting, because one might expect there to exist orbits with mean action close to the boundary rotation number. (Of course these always exist when the boundary rotation number is rational.)