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On the existence of Hamiltonian 4-manifolds with a contact type boundary

Aleksandra Marinkovic

Abstract

While the Hamiltonian group actions on closed symplectic manifolds have been widely explored throughout the last couple of decades, the study on Hamiltonian group actions on symplectic manifolds with a contact type boundary has started only recently, with the work by Niederkrüger and the author [MN]. In this note we pursue this study by presenting several methods to construct such Hamiltonian circle manifolds in dimension 4.

On the existence of Hamiltonian 4-manifolds with a contact type boundary

Abstract

While the Hamiltonian group actions on closed symplectic manifolds have been widely explored throughout the last couple of decades, the study on Hamiltonian group actions on symplectic manifolds with a contact type boundary has started only recently, with the work by Niederkrüger and the author [MN]. In this note we pursue this study by presenting several methods to construct such Hamiltonian circle manifolds in dimension 4.
Paper Structure (11 sections, 9 theorems, 68 equations, 1 figure)

This paper contains 11 sections, 9 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Properties of Hamiltonian manifolds with a contact type boundary
  3. Invariant collar neighborhood
  4. Legendrian orbits on the boundary of Hamiltonian 4-manifolds
  5. Basic examples
  6. Existence of a free Hamiltonian circle action
  7. Existence of critical surfaces
  8. Attaching a Weinstein Hamiltonian 2-handle
  9. Hamiltonian circle action on a Weinstein 2-handle.
  10. An invariant attaching map
  11. Blow up of an interior fixed point

Key Result

Lemma 2.2

For every boundary point $p\in V$ with $H(p) \neq0$ the gradient trajectory $\Phi_t^{\nabla H}(p)$ is non-constant. If $H(p)<0$ ($H(p)>0$) the trajectory $\Phi_t^{\nabla H}(p)$ starts (ends) in $p.$ Moreover, every trajectory can intersect the boundary only in end points.

Figures (1)

  • Figure 1: A model for a Weinstein 2-handle

Theorems & Definitions (29)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 19 more