A Partial Ordering on Slices of Planar Lagrangians

TL;DR

This paper introduces a partial order on slices of planar Lagrangian surfaces in three-dimensional space, exploring their structure and the relations defined by Lagrange cobordisms.

Contribution

It defines a new partial ordering on slices of planar Lagrangians and investigates the algebraic and geometric properties of this structure.

Findings

Negative slices are characterized and examples are provided.

Lagrange cobordisms induce a partial order on slices.

The set of slices forms a commutative monoid with an interesting additive relation.

Abstract

A collection of simple closed curves in $\rr^3$ is called a negative slice if it is the intersection of a flat-at-infinity planar Lagrangian surface and $\{y_2 = a \}$ for some $a < 0$. Examples and non-examples of negative slices are given. Embedded Lagrange cobordisms define a relation on slices and in some (and perhaps all) cases this relation defines a partial order. The set of slices is a commutative monoid and the additive structure has an interesting relationship with the ordering relation.