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.