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 be a compact domain in with smooth boundary . Assume that is “star-shaped”, meaning that is transverse to the radial vector field

.

Then the -form

restricts to a contact form on . Define to be the minimum symplectic action (period) of a Reeb orbit of . Define the *systolic ratio*

.

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

**Viterbo’s conjecture (weak version).** If is convex then .

**Example.** Consider the ellipsoid

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We have

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The boundary of the ellipsoid has Reeb orbits where for . The Reeb orbit has symplectic action

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(If any of the ratios for is rational, then there are additional Reeb orbits with higher symplectic action.) It follows that the systolic ratio

.

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 (even though the systolic ratio is). To get a symplectically invariant condition, define to be *dynamically convex* if every Reeb orbit on has Conley-Zehnder index at least . 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 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 with systolic ratio arbitrarily close to . (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 which give the tight contact structure and thus come from star-shaped hypersurfaces in . Let denote the unit disk, and let be the standard area form on , rescaled to have area ; in polar coordinates this is given by

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Also fix the primitive of defined by

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Now choose a real number . Let be an area-preserving diffeomorphism such that near the boundary, we have

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There is a unique function such that and near .

**Lemma.** Suppose that on all of . Then there is a contact form on giving the tight contact structure (in particular coming from a star-shaped hypersurface in ), compatible with an open book decomposition of with a page identified with , such that:

- ; in particular, the binding orbit has symplectic action .
- The return map of the Reeb flow from to itself is , and the return time is . In particular, Reeb orbits other than the binding correspond to periodic orbits of ; and the symplectic action of a periodic orbit is .
- .
- The binding orbit has rotation number with respect to a global trivialization of the contact structure, and thus Conley-Zehnder index when 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 -form on the complement of the binding, one starts with the -form on , and then glues together and via . This requires some modification of since and are not invariant under when is not a rotation. However we still have near the boundary of the disk where is a rotation. ABHS have a similar lemma with an alternate proof.

**Example.** Suppose that is simply rotation by on all of , not just near the boundary. Then , and we can construct the contact form as above without any correction. This contact form corresponds to the boundary of the ellipsoid . When is irrational, there are just two Reeb orbits: the binding, and the orbit coming from the fixed point of at the center of the disk. Note that while . (In general, the Euclidean volume of a four-dimensional star-shaped domain is half the contact volume of its boundary.)

**Example.** Suppose that , where is a monotone function such that for , and for . Then for , so the fixed point at the center has action approximately , and .

The idea of the ABHS construction is to combine the above two examples. Namely we will take , where is rotation by a rational number , and is as in the latter example, on each sector of the disk with , identified with a disk of area . To compute volume, action, and CZ indices for this example, we now introduce some definitions.

**Definition.** Inside a contact three-manifold , a *tube* of type , where and , is an invariant set for the Reeb flow, identified with , such that:

The contact form on is obtained by starting with the -form on , and then identifying with via rotation by . In particular, each disk has symplectic area ; the Reeb flow increases at speed ; and the return map is rotation by .

Moreover, the periodic orbit has rotation number with respect to a global trivialization of over (and thus Conley-Zehnder index when is irrational).

**Example.** In the boundary of the ellipsoid , the complement of the Reeb orbit of action is a tube of type .

Note that in general, a tube of type has contact volume .

Now suppose we have a tube of type where and are integers with and . We now introduce an operation which I will call “drilling” which reduces the volume of the tube. Let . The idea is to replace the return map by its composition with a map which rotates most of each of the sectors of backwards by . (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 . This new tube includes most of each of the sectors and wraps times around . The first (approximate) number is just the area of a sector. (The actual number is slightly less than this.) The second (approximate) number is times the original symplectic action , minus , similarly to the previous example. (The actual number is slightly greater than this.) The third (exact) number is times the original rotation number , minus because we rotated each sector by . Note that we have to assume that (as we had to assume before) to get a legitimate contact form.

In conclusion, the key formula to remember is

.

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

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

.

The upper limit of systolic ratios that we can get this way is what we would get if , which is

.

So by taking 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 will not have any Reeb orbits with as long as . So when we drill, to preserve dynamical convexity, we need to obtain a new tube with . One can argue that since , all new Reeb orbits created other than the Reeb orbit at the center of the new tube have larger Conley-Zehnder index, so is also sufficient to preserve dynamical convexity.

Now take in the above construction, so that we are starting with a tube of type . Then drilling by gives a tube of type . Dynamical convexity is preserved as long as . In the previous computation we saw that in this case we could take arbitrarily close to and obtain systolic ratio arbitrarily close to .

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 , and I suspect that it is impossible.