Book Review: From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds

Constructions of Lagrangian cobordisms

“…To formulate some precise motivating questions, we will use ribbon cobordism to denote a 2n-dimensional manifold that can be built from k-handles with k ≤ n. This idea of restricting the handle index is well known in symplectic topology: Eliashberg [CE12,Oan15] has shown that any 2n-dimensional Stein manifold admits a handle decomposition with handles of dimension at most n, and thus any 2n-dimensional Stein cobordism between closed, (2n−1)dimensional contact manifolds must be ribbon. Working in the relative setting with submanifolds and using the handle decomposition from the "height" function given by the R coordinate on R × R 3 , we see that all decomposable 2-dimensional Lagrangian cobordisms between 1-dimensional Legendrian submanifolds are ribbon cobordisms.…”