What is a Liouville Isomorphism in the Context of Symplectization?

[Kreizhn]

I shall use Seidel's definition of a Liouville domain; in particular, a Liouville domain is a compact manifold $M$ with boundary together with a one-form $\theta \in \Omega^1(M)$ such that $\omega = d\theta$ is a symplectic form and the vector field Z defined by $\iota_Z \omega = \theta$ is always strictly outward pointing along $\partial M$. Let $\alpha = \theta_{\partial M}$ be the contact form on the boundary.

Let $\hat M$ denote the symplectization of $M$ given by the natural collaring induced by the flow of Z. Namely, we attach an infinite cone to the boundary $\partial M$ and extend $\theta$ to $e^r \alpha$, and Z to $\frac\partial{\partial r}$ where $r$ is the $\mathbb (-\infty,0]$ coordinate of the symplectization.

Now a Liouville isomorphism of two Liouville domains $M_1, M_2$ is defined to be a diffeomorphism on the symplectizations $\hat M_1,\hat M_2$ such that $\phi^*\theta_2 = \theta_1 + dg$ where g is a compactly supported smooth function.

First question: Why do we define a Liouville isomorphism at the level of symplectization? Is the collaring argument really so canonical that it is essentially inherent to the definition of a Liouville domain?

Second question: Why do we only require that the one-form is preserved up to an exact form? I have heard an argument that demanding that the pullback preserve the form completely is too restrictive. Perhaps this could be elaborated upon. Furthermore, why do we need to the form to be exact? It seems to me that any argument about the preservation of structure could be done by using a compactly supported closed form, hence generalizing the space of Liouville isomorphisms even further.

 

[quasar987]

Well, I am not aware of the context of these definitions, but apparently, the reason for the definition of a Liouville iso is to generate symplectomorphisms of exact symplectic manifolds that are of the type "symplectification of Liouville domains". Indeed, take the exterior derivative of the defining formula for phi. So this class of symplectomorphism is in a sense simpler to identify than a general symplectomorphism since instead of asking for diffeos that preserve a two-form, we ask for those that preserve a 1-form... and not even that, we need only ask for those that preserve a 1-form up to an exact differential! Hopefully, this answers your second question?

As for question 1, yes the collaring construction is canonical. And it seems to me that a Liouville isomorphism between M1 and M2 has little to do with maps M1-->M2. Indeed, it is defined as a diffeomorphism btw the symplectifications which need not restrict to a diffeo M1-->M2! Mmmh, but I guess what you're really asking is if the map $M\mapsto \hat{M}$ is injective (up to symplectomorphism).