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Compactness of Moduli Spaces of Gradient Flow Lines in the Uniform Topology

Tom Stalljohann

Abstract

We prove a compactness result for gradient flow lines in a general set-up which comprises both the situation of Morse gradient flow lines as well as Floer cylinders converging to a critical submanifold respectively. For the compactness result we have to impose two conditions. Both are readily verified in the Morse case but establishing the second condition in the Floer case poses a technical challenge and relies on an exponential decay estimate for Floer cylinders, with coefficient function continuously depending on the initial loop. This is a result of independent interest.

Compactness of Moduli Spaces of Gradient Flow Lines in the Uniform Topology

Abstract

We prove a compactness result for gradient flow lines in a general set-up which comprises both the situation of Morse gradient flow lines as well as Floer cylinders converging to a critical submanifold respectively. For the compactness result we have to impose two conditions. Both are readily verified in the Morse case but establishing the second condition in the Floer case poses a technical challenge and relies on an exponential decay estimate for Floer cylinders, with coefficient function continuously depending on the initial loop. This is a result of independent interest.

Paper Structure

This paper contains 32 sections, 33 theorems, 191 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Compactness Result
  3. Motivation and Outlook
  4. Organization of the Paper
  5. Acknowledgments
  6. The Compactness Result for Moduli Spaces
  7. Uniform convergence and Coarse Metrics
  8. Conditions \ref{['it: ass A - Czeroloc convergence']} and \ref{['it: ass A - shortening gradient flow lines']}
  9. Sufficient Condition for \ref{['it: ass A - shortening gradient flow lines']}
  10. Proof of Theorem \ref{['mthm: Compactness moduli spaces']}
  11. Variants of the Compactness Theorem
  12. Illustrating the Compactness Result in the Morse Setting
  13. Gradient Flow Lines of Morse-Bott Functions
  14. Compactness of Moduli Spaces in an Ambient Banach Manifold
  15. Moduli Space of Unparametrized Trajectories
  16. ...and 17 more sections

Key Result

Theorem 1

Suppose $0 \leq E_0 < \mathfrak{S}^{+}_{f,G}(Z)$ and $I= [a,\infty)$ satisfy it: ass A - Czeroloc convergence and it: ass A - shortening gradient flow lines. Then for every sequence $(\gamma_n)_{n \in \mathbb{N}} \subseteq \mathcal{M}_{(E_0,I)}$ there exists a subsequence which converges in $C^0(I,M

Figures (2)

  • Figure 1: Illustrating the contradiction in the proof of Lemma \ref{['lem: uniform time for pt in Z']}. Two trajectories $\gamma_{n_1}$ and $\gamma_{n_2}$ for $n_1 \ll n_2$ are depicted. The entry points $\gamma_{n}(s_{n})$ into $V$ tend to $z_*$ while the end points $\gamma_n(\infty)$ tend to $z$. This contradicts assumption \ref{['it: ass A - shortening gradient flow lines']}, namely that the distance from $\gamma_n(s_n)$ to $\gamma_n(\infty)$ tends to zero.
  • Figure 2: Positive asymptotics for a trajectory $\gamma = \exp^{\Bar{g}}_{\gamma_0}(\widetilde{\xi}_{(\xi,v_-,v_+)})$ in the chart domain of $\Phi^{-1} \,$. The auxiliary metric $\Bar{g}$ is Euclidean on $(U_+,\varphi_+) \,$, so $\gamma(s) = \exp^{\Bar{g}}_{\gamma_0(s)}(\xi(s)) + \beta(s-s_0) \, v_+$ in the chart $\varphi_+$ due to \ref{['eq: Morse setting - exp gaux vs Euclidean exp']}.

Theorems & Definitions (90)

  • Definition 1.1: Spectral Gap
  • Definition 1.2
  • Definition 1.3: Class of Gradient Flow Lines
  • Remark 1.4
  • Example 1.5
  • Theorem 1
  • Corollary 2
  • Definition 2.1: Uniform convergence
  • Remark 2.2
  • Definition 2.3: Coarse Metric
  • ...and 80 more