Symplectic excision

TL;DR

This paper introduces a method using Hamiltonian flows to remove specific closed subsets from symplectic manifolds, addressing longstanding questions and generalizing previous results in symplectic topology.

Contribution

It develops a novel technique for symplectic excision of complex subsets, including Cantor-like sets and epigraphs, expanding the toolkit for symplectic topology.

Findings

Successfully excised a ray, answering Weinstein's question.

Generalized Stratmann's result to product spaces.

Provided examples like Cantor brushes and boxes with tails.

Abstract

We use time-independent incomplete Hamiltonian flows to excise interesting closed subsets of positive codimension from symplectic manifolds. Examples of such subsets include what we call a "Cantor brush", a "box with a tail", and -- more generally -- epigraphs of lower semicontinuous functions. This answers a question of Alan Weinstein about excision of a ray, and it generalizes a result of Bernd Stratmann about excision of the product of a ray with a manifold.