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Symplectic excision

Yael Karshon, Xiudi Tang

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.

Symplectic excision

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.

Paper Structure

This paper contains 4 sections, 4 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Excising a ray
  4. Excising a ray from Rn

Key Result

corollary 2.1

Let $N$ be a topological space, and let $D \subset N \times \mathbb{R}$ be a subset of the form for functions Then $D$ is open in $N \times \mathbb{R}$ if and only if $S$ is upper semi-continuous and $T$ is lower semi-continuous.

Theorems & Definitions (10)

  • proof
  • proof
  • corollary 2.1
  • proof
  • lemma 2.2
  • proof
  • lemma 2.3: Escape lemma, Version 1
  • proof
  • proposition 3.1
  • proof