Constructing exact Lagrangian immersions with few double points

TL;DR

This paper proves an h-principle for constructing exact Lagrangian immersions with minimal double points, showing that certain 3-manifolds can be immersed with only one double point in standard symplectic space.

Contribution

It establishes an h-principle for exact Lagrangian immersions with few double points and constructs explicit examples including a 3-manifold and a Lagrangian embedding with specific properties.

Findings

Any orientable closed 3-manifold admits an exact Lagrangian immersion with one double point.

Constructs a Lagrangian embedding of S^1×S^2 with vanishing Maslov class.

Provides a method to achieve minimal double points in Lagrangian immersions.

Abstract

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding $S^1\times S^2\to\R^6_\st$ with vanishing Maslov class.