Flexible Lagrangians

We introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed n-manifolds of dimension n > 2 can be realized as exact Lagrangian submanifolds of TSⁿ with possibly exotic Weinstein symplectic structures. These Weinstein structures on T Sⁿ, infinitely many of which are distinct, are formed by a single handle attachment to the standard 2n-ball along the Legendrian boundaries of flexible Lagrangians. We also formulate a number of open problems.