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On relative Hamiltonian diffeomorphisms

Ali Sait Demir

TL;DR

This work extends classical fragmentation techniques to the relative setting of Hamiltonian diffeomorphisms preserving a Lagrangian submanifold $L$, proving a Relative Fragmentation Theorem that decomposes any $h\in{\rm Ham}(M,L)$ into a product of compactly supported factors, with a refined version when $M$ is compact enforcing vanishing relative Calabi energy. It introduces and leverages key relative tools: the relative Calabi homomorphism, relative flux via Weinstein charts, and a fragmentation construction that localizes $h$ along a finite cover while controlling flux. The results build on Thurston and Banyaga’s methods, adapted to the constraint of preserving $L$, and reveal structural insights into ${\rm Ham}(M,L)$, including a discussion of why simplicity or perfectness may fail or remain open in the relative context. The paper thus lays a foundation for further study of the algebraic properties of relative Hamiltonian groups and highlights the limitations imposed by the relative setting on classical symmetry/group-theoretic arguments. The precise fragmentation mechanism and the interplay between relative flux, Calabi-type invariants, and Weinstein charts are the central contributions, with potential implications for understanding automorphism groups in symplectic geometry with boundary-like constraints.

Abstract

Let $\text{Ham(M,L)}$ denote the group of Hamiltonian diffeomorphisms on a symplectic manifold $M$, leaving a Lagrangian submanifold $L\subset M$ invariant. In this paper, we show that $\text{Ham(M,L)}$ has the fragmentation property, using relative versions of the techniques developed by Thurston and Banyaga.

On relative Hamiltonian diffeomorphisms

TL;DR

This work extends classical fragmentation techniques to the relative setting of Hamiltonian diffeomorphisms preserving a Lagrangian submanifold , proving a Relative Fragmentation Theorem that decomposes any into a product of compactly supported factors, with a refined version when is compact enforcing vanishing relative Calabi energy. It introduces and leverages key relative tools: the relative Calabi homomorphism, relative flux via Weinstein charts, and a fragmentation construction that localizes along a finite cover while controlling flux. The results build on Thurston and Banyaga’s methods, adapted to the constraint of preserving , and reveal structural insights into , including a discussion of why simplicity or perfectness may fail or remain open in the relative context. The paper thus lays a foundation for further study of the algebraic properties of relative Hamiltonian groups and highlights the limitations imposed by the relative setting on classical symmetry/group-theoretic arguments. The precise fragmentation mechanism and the interplay between relative flux, Calabi-type invariants, and Weinstein charts are the central contributions, with potential implications for understanding automorphism groups in symplectic geometry with boundary-like constraints.

Abstract

Let denote the group of Hamiltonian diffeomorphisms on a symplectic manifold , leaving a Lagrangian submanifold invariant. In this paper, we show that has the fragmentation property, using relative versions of the techniques developed by Thurston and Banyaga.
Paper Structure (6 sections, 5 theorems, 23 equations)

This paper contains 6 sections, 5 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Relative Hamiltonian Diffeomorphisms
  3. Relative Calabi Homomorphism
  4. Relative Weinstein Charts
  5. The Fragmentation Lemma
  6. Final Remarks

Key Result

Theorem 1.1

Let $\mathcal{U}=(U_j)_{j\in I}$ be an open cover of a compact, connected, symplectic manifold $(M,\omega)$ and $h$ be an element of ${\rm{Ham}}(M,L)$ for a Lagrangian submanifold $L$ of $M$. Then $h$ can be written where each $h_i\in {\rm{Ham}}_c(M,L), \ i=1,..,N$ is supported in $U_{j(i)}$ for some $j(i) \in I$. Moreover, if $M$ is compact, we may choose each $h_i$ such that $R_{U_i,U_i\cap L}(

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6