## The ABHS construction of contact forms with high systolic ratio

The purpose of this post, which accompanies a student seminar talk, is to explain a construction of Abbondandolo-Bramham-Hryniewicz-Salomão of contact forms with high systolic ratio.

To review what this means, let $X$ be a compact domain in ${\mathbb R}^{2n}$ with smooth boundary $Y$. Assume that $X$ is “star-shaped”, meaning that $Y$ is transverse to the radial vector field

$\rho = \frac{1}{2}\sum_{i=1}^n\left(x_i\frac{\partial}{\partial x_i} + y_i\frac{\partial}{\partial y_i}\right)$.

Then the $1$-form

$\lambda = \frac{1}{2} \sum_{i=1}^n\left(x_i\,dy_i - y_i\,dx_i\right)$

restricts to a contact form on $Y$. Define $c(X)$ to be the minimum symplectic action (period) of a Reeb orbit of $\lambda|_Y$. Define the systolic ratio

$\rho(X) = \frac{c(X)^n}{n!{\rm vol}(X)}$.

A famous conjecture, which I have been a bit obsessed with, is the following:

Viterbo’s conjecture (weak version). If $X$ is convex then $\rho(X)\le 1$.

Example. Consider the ellipsoid

$E(a_1,\ldots,a_n) = \left\{z\in{\mathbb C}^n\;\bigg|\;\sum_{i=1}^n\frac{\pi|z_i|^2}{a_i}\le 1\right\}$.

We have

${\rm vol}(E(a_1,\ldots,a_n)) = \frac{a_1\cdots a_n}{n!}$.

The boundary of the ellipsoid has Reeb orbits $\gamma_i$ where $z_j=0$ for $j\neq i$. The Reeb orbit $\gamma_i$ has symplectic action

${\mathcal A}(\gamma_i) = a_i$.

(If any of the ratios $a_i/a_j$ for $i\neq j$ is rational, then there are additional Reeb orbits with higher symplectic action.) It follows that the systolic ratio

$\rho(E(a_1,\ldots,a_n)) = \frac{\min(a_1,\ldots,a_n)^n}{a_1\cdots a_n}$.

In particular, we see that Viterbo’s conjecture holds for the ellipsoid, and is sharp for a ball.

The role of convexity in Viterbo’s conjecture is somewhat mysterious, since convexity is not invariant under symplectomorphisms between star-shaped domains in ${\mathbb R}^{2n}$ (even though the systolic ratio is). To get a symplectically invariant condition, define $X$ to be dynamically convex if every Reeb orbit on $Y$ has Conley-Zehnder index at least $n+1$. Hofer-Wysocki-Zehnder showed that convexity implies dynamical convexity. It is not known whether, conversely, every dynamically convex domain is symplectomorphic to a convex one.

The results of Abbondandolo-Bramham-Hryniewicz-Salamão which I want to explain are the following:

• (arXiv:1504.05258, Thm. 2) There exist star-shaped domains in ${\mathbb R}^4$ with arbitrarily high systolic ratio. (In particular the weak Viterbo conjecture does not hold for all star-shaped domains. This also disproves a conjecture I made on this blog a long time ago.)
• (arXiv:1710.06193, Thm. 1.1) There exist dynamically convex domains in ${\mathbb R}^4$ with systolic ratio arbitrarily close to $2$. (In particular it is not possible for both the weak Viterbo conjecture to be true and for every dynamically convex domain to be symplectomorphic to a convex one.)

To produce these examples, we first need to explain a general procedure for constructing contact forms on $S^3$ which give the tight contact structure and thus come from star-shaped hypersurfaces in ${\mathbb R}^4$. Let $D^2$ denote the unit disk, and let $\omega$ be the standard area form on $D^2$, rescaled to have area $A$; in polar coordinates this is given by

$\omega = \frac{Ar\,dr\,d\theta}{\pi}$.

Also fix the primitive of $\omega$ defined by

$\beta = \frac{Ar^2}{2\pi}d\theta$.

Now choose a real number $\theta_0>0$. Let $\phi:D^2\to D^2$ be an area-preserving diffeomorphism such that near the boundary, we have

$\phi(r,\theta) = (r,\theta + 2\pi\theta_0)$.

There is a unique function $f:D^2\to{\mathbb R}$ such that $df=\phi^*\beta-\beta$ and $f=\theta_0$ near $\partial D^2$.

Lemma. Suppose that $f>0$ on all of $D^2$. Then there is a contact form $\lambda$ on $S^3$ giving the tight contact structure (in particular coming from a star-shaped hypersurface in ${\mathbb R}^4$), compatible with an open book decomposition of $S^3$ with a page identified with $D^2$, such that:

• $d\lambda|_{D^2}=\omega$; in particular, the binding orbit has symplectic action $A$.
• The return map of the Reeb flow from $D^2$ to itself is $\phi$, and the return time is $f$. In particular, Reeb orbits other than the binding correspond to periodic orbits of $\phi$; and the symplectic action of a periodic orbit $(x_1,\ldots,x_d)$ is $\sum_{i=1}^df(x_i)$.
• ${\rm vol}(S^3,\lambda) = \int_{S^3}\lambda\wedge d\lambda = \int_{D^2}f\omega$.
• The binding orbit has rotation number $A/\theta_0+1$ with respect to a global trivialization of the contact structure, and thus Conley-Zehnder index $2\left\lfloor A/\theta_0\right\rfloor+3$ when $\theta_0$ is not an integer.

This lemma is from my paper “Mean action and the Calabi invariant”. The proof consists of a direct construction with differential forms. Roughly speaking, to obtain the $1$-form $\lambda$ on the complement of the binding, one starts with the $1$-form $\lambda = fdt+\beta$ on $[0,1]\times D^2$, and then glues together $\{0\}\times D^2$ and $\{1\}\times D^2$ via $\phi$. This requires some modification of $\lambda$ since $f$ and $\beta$ are not invariant under $\phi$ when $\phi$ is not a rotation. However we still have $\lambda = fdt+\beta$ near the boundary of the disk where $\phi$ is a rotation. ABHS have a similar lemma with an alternate proof.

Example. Suppose that $\phi$ is simply rotation by $\theta_0$ on all of $D^2$, not just near the boundary. Then $f\equiv\theta_0$, and we can construct the contact form as above without any correction. This contact form corresponds to the boundary of the ellipsoid $E(A,\theta_0)$. When $\theta_0$ is irrational, there are just two Reeb orbits: the binding, and the orbit coming from the fixed point of $\phi$ at the center of the disk. Note that ${\rm vol}(S^3,\lambda)=A\theta_0$ while ${\rm vol}(E(A,\theta_0))=A\theta_0/2$. (In general, the Euclidean volume of a four-dimensional star-shaped domain is half the contact volume of its boundary.)

Example. Suppose that $\phi(r,\theta)=(r,\theta+g(r))$, where $g$ is a monotone function such that $g(r)=\theta_0$ for $r>1-\epsilon$, and $g(r)=\theta_0-\psi$ for $r<1-2\epsilon$. Then $f\approx \theta_0-A\psi$ for $r<1-2\epsilon$, so the fixed point at the center has action approximately $\theta_0-A\psi$, and ${\rm vol}(S^3,\lambda)\approx A(\theta_0-A\psi)$.

The idea of the ABHS construction is to combine the above two examples. Namely we will take $\phi=\phi_1\circ\phi_2$, where $\phi_1$ is rotation by a rational number $p/q$, and $\phi_2$ is as in the latter example, on each sector of the disk with $2\pi k/q<\theta<2\pi(k+1)/q$, identified with a disk of area $1/q$. To compute volume, action, and CZ indices for this example, we now introduce some definitions.

Definition. Inside a contact three-manifold $(Y,\lambda)$, a tube of type $(A,L,\theta_0)$, where $A,L>0$ and $\theta_0\in{\mathbb R}$, is an invariant set for the Reeb flow, identified with $S^1\times D^2$, such that:

The contact form on $S^1\times D^2$ is obtained by starting with the $1$-form $\lambda = Ldt + Ar^2d\theta/(2\pi)$ on $[0,1]\times D^2$, and then identifying $\{0\}\times D^2$ with $\{1\}\times D^2$ via rotation by $2\pi\theta_0$. In particular, each disk $\{t\}\times D^2$ has symplectic area $A$; the Reeb flow increases $t$ at speed $1/L$; and the return map is rotation by $2\pi\theta_0$.

Moreover, the periodic orbit $S^1\times \{0\}$ has rotation number $\theta_0$ with respect to a global trivialization of $\xi$ over $S^3$ (and thus Conley-Zehnder index $2[\theta_0]+1$ when $\theta_0$ is irrational).

Example. In the boundary of the ellipsoid $E(a,b)$, the complement of the Reeb orbit of action $a$ is a tube of type $(a,b,b/a+1)$.

Note that in general, a tube of type $(A,L,\theta_0)$ has contact volume $AL$.

Now suppose we have a tube of type $(A,L,p/q)$ where $p$ and $q$ are integers with $q>0$ and ${\rm gcd}(p,q)=1$. We now introduce an operation which I will call “drilling” which reduces the volume of the tube. Let $\psi>0$. The idea is to replace the return map by its composition with a map which rotates most of each of the $q$ sectors of $D^2$ backwards by $\psi$. (This second map is the identity on a neighborhood of the “spokes” of the disk.) One can do this similarly to the lemma. The result contains a tube of type approximately $(A/q,qL-A\psi,p-q\psi)$. This new tube includes most of each of the $q$ sectors and wraps $q$ times around $S^1$. The first (approximate) number $A/q$ is just the area of a sector. (The actual number is slightly less than this.) The second (approximate) number is $q$ times the original symplectic action $L$, minus $A\psi$, similarly to the previous example. (The actual number is slightly greater than this.) The third (exact) number is $q$ times the original rotation number $p/q$, minus $q\psi$ because we rotated each sector by $-\psi$. Note that we have to assume that $qL-A\psi>0$ (as we had to assume $f>0$ before) to get a legitimate contact form.

In conclusion, the key formula to remember is

$(A,L,p/q) \longrightarrow (A/q,qL-A\psi,p-q\psi)$.

We can now play with this transformation to construct ABHS-type counterexamples.

Let’s start with the boundary of the ellipsoid $E(1,(q-1)/q)$, which contains a tube of type $(1,(q-1)/q,(2q-1)/q)$. If we drill by $\psi$, we obtain a tube of type approximately $(1/q,q-1-\psi,2q-1-q\psi)$. This tube, and also the entire new contact manifold, has contact volume approximately $(q-1-\psi)/q$. What about the Reeb orbits? The binding of the ellipsoid (which we deleted to obtain the tube) has action $1$, the orbit at the center of the old tube (which is still there) has action $(q-1)/q$, and the orbit at the center of the new tube has action approximately $q-1-\psi$. One can argue that all other Reeb orbits have action at least as big as one of the preceding ones. As a result, if $\psi<(q-1)^2/q$, so that the orbit at the center of the old tube is the shortest one, then the systolic ratio is approximately

$\rho \approx \frac{((q-1)/q)^2}{(q-1-\psi)/q} = \frac{(q-1)^2}{q(q-1-\psi)}$.

The upper limit of systolic ratios that we can get this way is what we would get if $\psi=(q-1)^2/q$, which is

$\rho \approx \frac{(q-1)^2}{q(q-1-(q-1)^2/q)} = q-1$.

So by taking $q$ to be an arbitrarily large integer, we can obtain an arbitrarily large systolic ratio!

What if we also want dynamical convexity? A tube of type $(A,L,\theta_0)$ will not have any Reeb orbits with $CZ<3$ as long as $\theta_0>1$. So when we drill, to preserve dynamical convexity, we need to obtain a new tube with $\theta_0>1$. One can argue that since $\psi>0$, all new Reeb orbits created other than the Reeb orbit at the center of the new tube have larger Conley-Zehnder index, so $\theta_0>1$ is also sufficient to preserve dynamical convexity.

Now take $q=3$ in the above construction, so that we are starting with a tube of type $(1,2/3,5/3)$. Then drilling by $\psi$ gives a tube of type $(1/3,2-\psi,5-3\psi)$. Dynamical convexity is preserved as long as $\psi<4/3$. In the previous computation we saw that in this case we could take $\psi$ arbitrarily close to $4/3$ and obtain systolic ratio arbitrarily close to $2$.

One can try plugging other numbers into this construction, and one can also iterate the drilling operation, but I haven’t been able to obtain a dynamically convex example this way with systolic ratio greater than $2$, and I suspect that it is impossible.