Convex boundary of a symplectic manifold

Solution 1:

I found the following in "Convex Symplectic Manifolds" by Eliashberg and Gromov; you can find this article in Several complex variables and complex geometry (Proceedings of symposia in pure mathematics ; v. 52, pt. 2). I've translated it into different notation.

Let $S$ be a hypersurface of contact type bounding a symplectic manifold $M$. We say $S$ is convex if there is a Liouville vector field near $S$ that points outwards, and concave if there is one that points inwards.

The case of a surface with boundary $S^1$ is given as - and, as above, is - an example of a boundary that is both convex and concave.

Now let $\alpha$ be a compatible contact structure on $S$ - that is, that $d\alpha = \omega$ restricted to $S$. Call the Reeb field for this contact form $\delta$. Gromov says that if $G$ is a null-homologous closed Reeb orbit for $\alpha$, then $\int_G \alpha' > 0$ for all compatible contact forms $\alpha'$. This implies that $S$ cannot be simultaneously convex and concave. (From Liouville fields pointing in opposite directions, we can produce contact forms, compatible with $\omega$, that are homologically negative of each other.)

In particular, consider $S^3$ as the boundary of your favorite symplectic manifold with boundary $S^3$. Then it cannot be both convex and concave. In addition, there is a symplectically fillable contact structure on $T^3$ with no null-homologous Reeb orbits; so this does not provide an obstruction to being both convex and concave. In fact, it's both with respect to a certain symplectic filling. (Note that every closed 3-manifold with a given contact form has a closed Reeb orbit; this was proved by Taubes in 2007, and is a special case of the Weinstein conjecture.)

As a result of this, I don't understand the exercise in McDuff and Salamon. If I interpret it to say "Given a hypersurface of contact type (without a chosen orientation), every transverse Liouville vector field near this hypersurface points in the same direction", it's false. If I've already given it an orientation, I don't see how the question isn't trivial.