## Is there a quantitative refinement of the Weinstein conjecture?

The following post consists of various conjectures and speculations which seem hard to prove if true, although maybe not so hard to disprove in the likely event that they are false.  I am hoping some reader might be able to quickly shoot down some of these conjectures, or have some ideas or know some references for positive results.

Recall that the three-dimensional Weinstein conjecture, proved in full generality by Taubes, asserts that for any closed oriented three-manifold $Y$ with a contact form $\lambda$, there exists a Reeb orbit.

Question: does there exist a Reeb orbit with an explicit upper bound on the length?

To make this question more precise, define $\ell(Y,\lambda)$ to be the length of the shortest Reeb orbit.  Here the “length”, or “symplectic action”, of a Reeb orbit is defined to be the integral of $\lambda$ over it.  We would like an explicit upper bound on $\ell(Y,\lambda)$.  Note that this is impossible without fixing the volume or something, because if we scale the contact form by multiplying it by a positive constant $r$, then $\ell(Y,r\lambda)=r\cdot\ell(Y,\lambda)$.  We can get a scale-invariant quantity by taking the quotient

$c(Y,\lambda) = \frac{\ell(Y,\lambda)}{\sqrt{vol(Y,\lambda)}}.$

Here $vol(Y,\lambda)=\int_Y\lambda\wedge d\lambda$.  Now if $\xi$ is a contact structure on $Y$, define

$c(Y,\xi) = \sup\{c(Y,\lambda) \mid Ker(\lambda)=\xi\}.$

Question: Is $c(Y,\xi) < \infty$?  If so, do the numbers $c(Y,\xi)$ have a universal upper bound?  And to what extent do these numbers distinguish contact structures?

The numbers $c(Y,\xi)$ might be impossible to compute.  Let me still make:

Wild speculation: $c(Y,\xi) \le \sqrt{2}$ for all $(Y,\xi)$.

This is suggested by the theory of ECH capacities for reasons discussed here.

Example 1: Let $\Sigma$ be a closed connected surface and let $Y$ be a principal $S^1$ bundle over $\Sigma$ with euler class $e> 0$.  Let $\lambda$ be a connection $1$-form on this circle bundle with everywhere nonvanishing curvature.  Then $\lambda$ is a contact form, and the Reeb flow is given by the $S^1$ action that rotates the fibers.  In particular $\ell(Y,\lambda)=2\pi$.  On the other hand $vol(Y,\lambda) = 4\pi^2 e$.  Thus $c(Y,\lambda) = 1/\sqrt{e}$.

It is tempting to speculate that if $\xi$ denotes the contact structure in the above example, then $c(Y,\xi) = 1/\sqrt{e}$.  It is obvious that $c(Y,\lambda)$ decreases if we multiply the above $\lambda$ by a nonconstant $S^1$-invariant function.  However this speculation is disproved by Example 2 below when $\Sigma = S^2$ and $e=2$.

Example 2: Let $\Sigma$ be a closed connected surface, and let $Y$ be the unit cotangent bundle of $\Sigma$ with the canonical contact form $\lambda$ for which the Reeb flow is the geodesic flow.  Then $\ell(Y,\lambda)$ is the length of the shortest geodesic on $\Sigma$, while $vol(Y,\lambda) = 2\pi A$, where $A$ denotes the area of $\Sigma$. Now there is lots of literature giving upper bounds on $\ell^2/A$ which depend only on the genus of $\Sigma$.  In the case $\Sigma = S^2$, I found a recent paper by Regina Rotman proving that $\ell^2/A \le 32$.  It was previously conjectured by Calabi and Croke that $\ell^2/A \le 2\sqrt{3}$.   (Explicit, non-symmetric examples show that no better upper bound is possible.)  Note that my first wild speculation above would imply that $\ell^2/A \le 4\pi$, which interestingly is between the known upper bound and the conjectured upper bound.  For higher genus orientable surfaces, according to the Wikipedia article on “systoles of surfaces” (I didn’t check the original sources), it is known that $\ell^2/A \le 2/\sqrt{3}$ for genus $1$, $2$, and $\ge 20$, with the inequalities realized by noncontractible geodesics.  I don’t know whether the literature can confirm or deny my wild speculation when the genus is between $2$ and $20$.

Example 3: Consider the three-torus $Y=({\mathbb R}^2/{\mathbb Z}^2)^3$ with coordinates $x,y,z$.  Take $\lambda = f(z)dx + g(z)dy$ where $(f,g)$, as a function of $z$, describes a plane curve whose argument ($\theta$ in polar coordinates) is a strictly increasing function of $z$.  This is a contact form whose volume is twice the area encosed by the curve $(f,g)$.  Suppose for simplicity that the curve $(f,g)$ is perpendicular to the coordinate axes.  Then $\ell(Y,\lambda)$ is at most the minimum distance to the origin of an intersection of $(f,g)$ with a coordinate axis.  So for these examples $c(Y,\lambda) \le \sqrt{vol(Y,\lambda)/n}$, where $n$ denotes the winding number of the curve $(f,g)$ around the origin.  Note that up to diffeomorphism, the contact structure depends only on $n$; let us denote it by $\xi_n$.  The above picture suggests:

Conjecture: $c(Y,\xi_n)=c(Y,\xi_1)/\sqrt{n}$.

In particular, if these numbers $c$ are finite as hoped, then they can distinguish contact structures on the same three-manifold.

Example 4: Open books.  This might be a promising place to seek counterexamples to the above conjectures, but I don’t understand the situation here very well.