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.

 

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