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Stable Morse flow trees

Kenneth Blakey

TL;DR

The paper proves a Floer-Gromov-type compactness theorem for (pre)stable Morse flow trees of Legendrians in 1-jet spaces with simple front singularities, under Ekholm's preliminary transversality condition. It develops a framework of prestable and stable Morse flow trees, introduces pre-Floer-Gromov and Floer-Gromov convergence, and defines a unique compact topology on the moduli space $\overline{\mathfrak{S}}(L,N)$ by combining edge-convergence with degeneration phenomena such as ghost edges and nodal vertices. The results rely on a Morse-theoretic compactification of edge moduli, a Boardman–Vogt/associahedron structure for the domain trees, and a detailed analysis of gradient flows of local function differences $f_i-f_j$ between sheets of $L$. The construction provides a robust framework for Legendrian contact homology computations in $1$-jet spaces and clarifies degeneration patterns beyond rigid (1-puncture) configurations, with potential implications for multiply-covered configurations and nodal degenerations in the Floer-theoretic setting.

Abstract

We prove a Floer-Gromov compactness type result for (stable) Morse flow trees of Legendrians in 1-jet spaces with simple front singularities satisfying Ekholm's preliminary transversality condition.

Stable Morse flow trees

TL;DR

The paper proves a Floer-Gromov-type compactness theorem for (pre)stable Morse flow trees of Legendrians in 1-jet spaces with simple front singularities, under Ekholm's preliminary transversality condition. It develops a framework of prestable and stable Morse flow trees, introduces pre-Floer-Gromov and Floer-Gromov convergence, and defines a unique compact topology on the moduli space by combining edge-convergence with degeneration phenomena such as ghost edges and nodal vertices. The results rely on a Morse-theoretic compactification of edge moduli, a Boardman–Vogt/associahedron structure for the domain trees, and a detailed analysis of gradient flows of local function differences between sheets of . The construction provides a robust framework for Legendrian contact homology computations in -jet spaces and clarifies degeneration patterns beyond rigid (1-puncture) configurations, with potential implications for multiply-covered configurations and nodal degenerations in the Floer-theoretic setting.

Abstract

We prove a Floer-Gromov compactness type result for (stable) Morse flow trees of Legendrians in 1-jet spaces with simple front singularities satisfying Ekholm's preliminary transversality condition.
Paper Structure (17 sections, 12 theorems, 89 equations, 4 figures, 1 table)

This paper contains 17 sections, 12 theorems, 89 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Context
  3. Main results
  4. Preliminaries on Morse flow trees
  5. Metric ribbon trees
  6. Local function differences
  7. Morse flow trees
  8. Compactifying edges
  9. Morse theory preliminaries
  10. Edge convergence
  11. Notation
  12. Topologies
  13. Compactness
  14. Compactifying stable Morse flow trees
  15. Stable broken Morse flow trees
  16. ...and 2 more sections

Key Result

Theorem 1.1

There exists a topology on $\overline{\mathfrak{S}}(L,N)$ which is compact; moreover, this topology is unique with respect to the property that a sequence topologically converges if and only if that sequence Floer-Gromov converges.

Figures (4)

  • Figure 1: (Figure 17 in Ekh07) (Top) The top view of $F_1'$. (Bottom) The cross-sectional view of $F_1'$.
  • Figure 2: (Figure 18 in Ekh07) The sequence of stable Morse flow trees $\{T_\nu\}$.
  • Figure 3: (Figure 19 in Ekh07) The prestable Morse flow tree $T^\mathrm{pre}$.
  • Figure 4: (Figure 20 in Ekh07) The sequence of stable Morse flow trees $\{T_\nu'\}$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7: Lemma 2.8 in Ekh07
  • Definition 2.8
  • ...and 38 more