**Questions**:

(1) Can we improve lower bound on number of closed Reeb orbits?

(2) Can we get existence with a priori upper bound on length?

(3) Can we replace Reeb vector fields with something else?

**Theorem** for (1): Hutchings & Taubes. 2008. $M^3$ contact, compact, oriented. At least two distinct orbits for non-degenerate. If manifold is not $S^3$ or lens space, then have at least 3 (again, assume non-degenerate).

**Theorem** (Cristofaro-Gardiner & Gripp-Ramos 2012)

Every possibly degenerate contact form on a closed oriented $M^3$ has two embedded Reeb orbits.

**Conjecture** (Ekeland-Hofer): every star-shaped hypersurface in $\mathbb{R}^{2n}$ has at least $n$ closed Reeb orbits.

If contact form non-degenerate, conjecture follows by an argument from CH algebra. (Jean Gutt explained this to me earlier. [INSERT EXPLANATION HERE])

Colin-Honda 2008: for many contact structures $\xi$ on closed oriented $Y^3$, every contact form $\ker \alpha = \xi$ has infinitely many distinct orbits. (Uses linearized CH.)

**Conjecture**: $Y^3$ closed oriented connected different from $S^3$ or a lens space, then every contact form on $Y$ has infinitely many embedded Reeb orbits.

**Definition**: volume of $(Y, \alpha)$ is $\int \alpha \wedge d\alpha$.

**Question: **Does there exist $c(Y, \xi)$ so $A(\gamma)^2 \le c(Y, \xi) vol(Y, \alpha)$ for some closed $\gamma$?

**Stronger version**: $c(Y, \xi) \le 1$.

$\newcommand{\vol}{\mathrm{vol}}$

**Examples:** Take a prequantization bundle with Euler class $e$ and $\omega = 2\pi e$. Take some positive function $f \colon \Sigma \to \mathbb{R}$. Then look at $f \alpha_0$.

Conjecture holds for this example, since $ vol( Y, f \alpha_0) = 2 \pi \int_{\Sigma} f^2 \omega

\ge 4 \pi^2 e \min(f^2)$. Action of critical fibres are $2\pi f(x)$ (where $x$ is a critical point of $f$.)

Another example: relationship to systolic inequalities.

Take $g$ a Riemannian metric on $S^2$ with area $4 \pi$. Then, action of a Reeb orbit agrees with the length of the corresponding geodesic.

Go to double cover $S^3$ and take pull-back contact form. This gives volume $16 \pi^2$. Reeb orbits are even geoedeics (geods which determine plane curves with odd rotation number). The conjecture then says there is an even geodesic of length $\le 4 \pi$.

This is sharp for the round metric (double cover of a great circle on $S^2$).

Calabi-Croke example: two flat equilateral triangles glued together to give metric of area $4 \pi$ and no closed geodesics of length $\le 2 \pi$. However! exists an even geodesic of length $\le 4 \pi$.

**Sketch of proof **: suppose $Y$ has only two Reeb orbits. Then, some trickery and work shows that you can construct a foliation from $u$ map. (Foliation by cylinders) This gives a genus 1 Heegaard splitting.

**Some consequences of isomorphism to SW Floer homology**:

non-canonical $\mathbb{Z}$ grading when $c_1(\xi) + 2 PD(\Gamma)$ is torsion.

For sufficiently negative grading, $ECH$ vanishes. For sufficiently large grading, $U : ECH_* \to ECH_{*-2}$ is isomorphism. Infinitely many $*$ so $ECH_*$ non-zero.

Filtered ECH: look at subcomplex filtered by action. This depends heavily on the contact form.

Michael with Daniel & Vinicius (2012) : consider sequence of classes so that $U \sigma_{k+1} = \sigma_k$. Then they show $\lim_k c_{\sigma_k}(Y, \alpha)^2/ k = 2 \vol(Y, \alpha)$.

Another theorem with the usual suspects: Suppose $(Y, \alpha)$ closed connected 3-man. Then, either exists a short Reeb orbit OR at least 3 embedded Reeb orbits.

Idea uses some trickery with the $U$ map. I guess the key idea is that having 2 orbits means that the picture looks an awful lot like the ellipsoid.