Flexible Lagrangians

Abstract

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 $T^{\ast }{S}^n$ with possibly exotic Weinstein symplectic structures. These Weinstein structures on $T^{\ast }{S}^n$, 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.

arXiv comment14 pages, 3 figures New version is restructured and expanded, new results are added