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Symplectic excision and distance rigidity

Yoel Groman

TL;DR

The paper investigates how completeness properties behave under excision in open symplectic manifolds, introducing a strong rigidity phenomenon when the excised set is a symplectic hypersurface via open Gromov-Witten invariants, contrasted with flexibility in coisotropic excisions. It develops two complementary completeness notions: geometric boundedness and normalized completeness, the latter built from an isoperimetric scale ρ extsubscript{J} and yielding a normalized distance d extsubscript{norm,J} that is robust under C extsuperscript{0} perturbations and under uniform geometric bounds. Core results include that removing a symplectic hypersurface destroys geometric boundedness (tm:1), and, in a toric/convex setting, flux-distance controls boundary separation (tm:2); obstructions to existence of normalized complete structures in Tot(L) and Tot(O(D)) are also established (tm:3, tm:4). The normalized framework aligns with Fukaya category ideas, suggesting a link between geometric completeness and moduli-space completeness of objects, while providing concrete tools (confinement lemmas, isoperimetric scales) applicable to symplectic topology and mirror-symmetric contexts. Overall, the work delineates rigidity and flexibility regimes for completeness under symplectic excision and refines a C extsuperscript{0}-stable, Fukaya-category-inspired notion of completeness with precise quantitative bounds.

Abstract

We consider various notions of completeness in symplectic topology and ask two related questions. Does a complete open symplectic manifold remain complete after excising a subset? Can two sets be made arbitrarily far apart by adjusting the almost complex structure within an appropriate class of complete almost complex structures? We find rigidity phenomena when the excised set is a symplectic hypersurface. These arise from certain open Gromov-Witten invariants. We contrast this with flexibility that often occurs when the excised set is coisotropic. We also briefly touch on the opposite question of obstructions to existence of a complete symplectic structure compatible with a given complex structure. For the notion of completeness we first consider the traditional notion of geometric boundedness. We then introduce a broader notion of normalized completeness, related to the notion of intermittent boundedness of \cite{GromanFloerOpen}, which depends on $C^0$ properties and is a contractible condition. Finally we speculate about the relation to a Fukaya-categorical notion of completeness.

Symplectic excision and distance rigidity

TL;DR

The paper investigates how completeness properties behave under excision in open symplectic manifolds, introducing a strong rigidity phenomenon when the excised set is a symplectic hypersurface via open Gromov-Witten invariants, contrasted with flexibility in coisotropic excisions. It develops two complementary completeness notions: geometric boundedness and normalized completeness, the latter built from an isoperimetric scale ρ extsubscript{J} and yielding a normalized distance d extsubscript{norm,J} that is robust under C extsuperscript{0} perturbations and under uniform geometric bounds. Core results include that removing a symplectic hypersurface destroys geometric boundedness (tm:1), and, in a toric/convex setting, flux-distance controls boundary separation (tm:2); obstructions to existence of normalized complete structures in Tot(L) and Tot(O(D)) are also established (tm:3, tm:4). The normalized framework aligns with Fukaya category ideas, suggesting a link between geometric completeness and moduli-space completeness of objects, while providing concrete tools (confinement lemmas, isoperimetric scales) applicable to symplectic topology and mirror-symmetric contexts. Overall, the work delineates rigidity and flexibility regimes for completeness under symplectic excision and refines a C extsuperscript{0}-stable, Fukaya-category-inspired notion of completeness with precise quantitative bounds.

Abstract

We consider various notions of completeness in symplectic topology and ask two related questions. Does a complete open symplectic manifold remain complete after excising a subset? Can two sets be made arbitrarily far apart by adjusting the almost complex structure within an appropriate class of complete almost complex structures? We find rigidity phenomena when the excised set is a symplectic hypersurface. These arise from certain open Gromov-Witten invariants. We contrast this with flexibility that often occurs when the excised set is coisotropic. We also briefly touch on the opposite question of obstructions to existence of a complete symplectic structure compatible with a given complex structure. For the notion of completeness we first consider the traditional notion of geometric boundedness. We then introduce a broader notion of normalized completeness, related to the notion of intermittent boundedness of \cite{GromanFloerOpen}, which depends on properties and is a contractible condition. Finally we speculate about the relation to a Fukaya-categorical notion of completeness.
Paper Structure (21 sections, 18 theorems, 51 equations)

This paper contains 21 sections, 18 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. The results
  3. Flexibility of coisotropic excision
  4. A basic 4-dimensional example.
  5. Holomorphic-curve mechanism and the surface case
  6. Completeness in symplectic topology
  7. Fukaya-categorical completeness
  8. Proofs
  9. Energy confinement for $J$-holomorphic curves
  10. Proof of Theorem \ref{['tm:1']}
  11. Proof of Theorem \ref{['tm:2']}
  12. Proof of Theorems \ref{['tm:3']} and \ref{['tm:4']}
  13. Normalized completeness
  14. An isoperimetric scale
  15. Normalized length and normalized completeness
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $M$ be a geometrically bounded symplectic manifold with finitely generated spherical homology, and suppose $\omega$ is rational. Let $N\subset M$ be a properly embedded symplectic hypersurface. Then $M\setminus N$ is not geometrically bounded.

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 1.9: $J$-completeness
  • Definition 1.10: Normalized completeness
  • ...and 48 more