Flexible Lagrangians

TL;DR

This paper explores the concepts of regularity and flexibility in Lagrangian manifolds with Legendrian boundary within Weinstein domains, revealing abundant flexible Lagrangians and new constructions of Legendrian and Weinstein manifolds.

Contribution

It introduces notions of regularity and flexibility for Lagrangians with Legendrian boundary, leading to novel constructions and examples of Weinstein structures and Legendrian submanifolds.

Findings

Many closed n-manifolds can be realized as exact Lagrangians in T* S^n.

Existence of infinitely many distinct Weinstein structures on T* S^n.

Flexible Lagrangians enable new constructions of Legendrian and Weinstein manifolds.

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 $T^*S^n$ with possibly exotic Weinstein symplectic structures. These Weinstein structures on $T^* S^n$, 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.