# Reference for plumbing cotangent bundles as Liouville manifolds

A Liouville domain $(W, \omega,\alpha, X)$ is a compact manifold $W$ with boundary $\partial W$, and a exact symplectic structure $\omega = d\alpha, \iota_X \omega = \alpha$, such that $X$ points outward along $\partial W$.

The simplest example is unit disk bundle in the cotangent bundle of a compact Riemannian manifold $M$: $$W = D^*M = \{(q,p) \in T^*M \mid \|p\| \leq 1 \}$$ with the canonical one-form $\alpha = \sum_i p_i dq_i$ and the expanding Liouville field $X = \sum_i p_i \partial_{p_i}$.

Given two smooth manifolds $L_1,L_2$ of dimension $n$, and fix base points $q_i \in L_i$, we may define a symplectic connected sum $T^* L_1 \#_{q_1 \sim q_2} T^*L_2$, by identifying the neighborhoods of $(q_i,0) \in T^*L_i$ via $$T^* \mathbb{R}^n \ni (x,y) \mapsto (-y, x) \in T^* \mathbb{R}^n,$$ such that $L_1,L_2$ intersects as the zero-section and a cotangent fiber of $T^* \mathbb{R}^n$.

It seems to be a standard fact that the plumbing of cotangent bundles admit a Liouville structure, see eg here (page 5-6). However, I can only see how to obtain Liouville structure locally, in open sets either includes $q$ or excludes $q$, but not sure how to glue them together. Any references for an explicit construction would be much appreciated. Thanks!