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Examples of symplectic non-leaves

Fabio Gironella, Lauran Toussaint

TL;DR

The paper addresses which open manifolds can occur as leaves of codimension-1 symplectic foliations on closed manifolds, showing that leaves must be strongly geometrically bounded (sgb) and that an exhaustion by convex-contact-type compacts allows deformation to an sgb form. It then proves topological and volumetric obstructions that rule out many manifolds from being diffeomorphic to symplectic leaves, including $ obreak R^{2n}$ as a proper leaf, while still permitting smooth leaf realizations or dense leaves in suitable foliations. Finally, it constructs explicit non-leaves via symplectic blowups at infinitely many points—with varying exceptional volumes—to produce sgb forms not symplectomorphic to leaves, and it demonstrates both leaf-diffeomorphic and non-leaf scenarios in blowup settings, highlighting a clear distinction between smooth leafability and symplectic leafability. These results illuminate the gap between smooth foliations and symplectic foliations and provide concrete counterexamples, as well as techniques (blowups, contact-type exhaustions) for generating non-leaves in the symplectic category.

Abstract

This paper deals with the following question: which manifolds can be realized as leaves of codimension-1 symplectic foliations on closed manifolds? We first observe that leaves of symplectic foliations are necessarily strongly geometrically bounded. We show that a symplectic structure which admits an exhaustion by compacts with (convex) contact boundary can be deformed to a strongly geometrically bounded one. We then give examples of smooth manifolds which admit a strongly geometrically bounded symplectic form and can be realized as a smooth leaf, but not as a symplectic leaf for any choice of symplectic form on them. Lastly, we show that the (complex) blowup of 2n-dimensional Euclidean space at infinitely many points, both admits strongly geometrically bounded symplectic forms for which it can and cannot be realized as a symplectic leaf.

Examples of symplectic non-leaves

TL;DR

The paper addresses which open manifolds can occur as leaves of codimension-1 symplectic foliations on closed manifolds, showing that leaves must be strongly geometrically bounded (sgb) and that an exhaustion by convex-contact-type compacts allows deformation to an sgb form. It then proves topological and volumetric obstructions that rule out many manifolds from being diffeomorphic to symplectic leaves, including as a proper leaf, while still permitting smooth leaf realizations or dense leaves in suitable foliations. Finally, it constructs explicit non-leaves via symplectic blowups at infinitely many points—with varying exceptional volumes—to produce sgb forms not symplectomorphic to leaves, and it demonstrates both leaf-diffeomorphic and non-leaf scenarios in blowup settings, highlighting a clear distinction between smooth leafability and symplectic leafability. These results illuminate the gap between smooth foliations and symplectic foliations and provide concrete counterexamples, as well as techniques (blowups, contact-type exhaustions) for generating non-leaves in the symplectic category.

Abstract

This paper deals with the following question: which manifolds can be realized as leaves of codimension-1 symplectic foliations on closed manifolds? We first observe that leaves of symplectic foliations are necessarily strongly geometrically bounded. We show that a symplectic structure which admits an exhaustion by compacts with (convex) contact boundary can be deformed to a strongly geometrically bounded one. We then give examples of smooth manifolds which admit a strongly geometrically bounded symplectic form and can be realized as a smooth leaf, but not as a symplectic leaf for any choice of symplectic form on them. Lastly, we show that the (complex) blowup of 2n-dimensional Euclidean space at infinitely many points, both admits strongly geometrically bounded symplectic forms for which it can and cannot be realized as a symplectic leaf.

Paper Structure

This paper contains 4 sections, 17 theorems, 29 equations.

Table of Contents

  1. Ends of manifolds and accumulation of leaves
  2. Geometrically bounded symplectic manifolds
  3. Manifolds not diffeomorphic to symplectic leaves
  4. Examples of symplectic non-leaves via blowups

Key Result

Theorem 1

Let $(W,\omega)$ be a symplectic manifold admitting an exhaustion $\mathcal{K}$ of contact type. Then $\omega$ is homotopic through symplectic forms to a sgb symplectic form $\omega'$.

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Corollary 6
  • Theorem 7: CanCon00
  • Definition 8
  • Proposition 9
  • proof
  • ...and 23 more