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Local gluing

Urs Frauenfelder, Joa Weber

Abstract

In the local gluing one glues local neighborhoods around the critical point of the stable and unstable manifolds to gradient flow lines defined on a finite time interval $[-T,T]$ for large $T$. If the Riemannian metric around the critical point is locally Euclidean, the local gluing map can be written down explicitly. In the non-Euclidean case the construction of the local gluing map requires an intricate version of the implicit function theorem. In this paper we explain a functional analytic approach how the local gluing map can be defined. For that we are working on infinite dimensional path spaces and also interpret stable and unstable manifolds as submanifolds of path spaces. The advantage of this approach is that similar functional analytical techniques can as well be generalized to infinite dimensional versions of Morse theory, for example Floer theory. A crucial ingredient is the Newton-Picard map. We work out an abstract version of it which does not involve troublesome quadratic estimates.

Local gluing

Abstract

In the local gluing one glues local neighborhoods around the critical point of the stable and unstable manifolds to gradient flow lines defined on a finite time interval for large . If the Riemannian metric around the critical point is locally Euclidean, the local gluing map can be written down explicitly. In the non-Euclidean case the construction of the local gluing map requires an intricate version of the implicit function theorem. In this paper we explain a functional analytic approach how the local gluing map can be defined. For that we are working on infinite dimensional path spaces and also interpret stable and unstable manifolds as submanifolds of path spaces. The advantage of this approach is that similar functional analytical techniques can as well be generalized to infinite dimensional versions of Morse theory, for example Floer theory. A crucial ingredient is the Newton-Picard map. We work out an abstract version of it which does not involve troublesome quadratic estimates.
Paper Structure (21 sections, 31 theorems, 232 equations, 3 figures)

This paper contains 21 sections, 31 theorems, 232 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Local gluing map for the Euclidean metric
  3. Local gluing map for a general Riemannian metric
  4. Setup -- path spaces and sections
  5. Idea of proof
  6. Pre-gluing map ${\mathcal{P}}_T$ and its restriction $\wp_T$
  7. Infinitesimal gluing
  8. Projection associated to a particular complement
  9. Infinitesimal gluing map $\Gamma_T$
  10. Newton-Picard map
  11. Approximate zero
  12. Surjectivity and right inverse
  13. Definition of ${\mathcal{N}}_T$
  14. Gluing
  15. Diffeomorphism onto image
  16. ...and 6 more sections

Key Result

Theorem A

There are open neighborhoods ${\mathcal{U}}_+$ and ${\mathcal{U}}_-$ of the origin in the stable and unstable manifold and gluing maps $\gamma_T\colon {\mathcal{U}}_+\times{\mathcal{U}}_-\to{\mathcal{M}}_T$ for $T\ge T_0$, where ${\mathcal{M}}_T$ is the space of downward gradient flow lines on the f

Figures (3)

  • Figure 1: Convergence of local gluing ${\rm ev}_T\circ\gamma_T(w_+,w_-) \stackrel{T\to\infty}{\longrightarrow}{\rm ev}(w_+,w_-)$
  • Figure 2: Pre-glued path $w_T(s):={\mathcal{P}}_T(w_+,w_-)(s)$ for $s\in[-T,T]$
  • Figure 3: Infinitesimal gluing isomorphism $\Gamma_T\in{\mathcal{L}}({\mathbb{E}}^+\times{\mathbb{E}}^-,{\mathbb{E}}_T)$; cf. (\ref{['eq:Gamma_T']})

Theorems & Definitions (73)

  • Theorem A: Local gluing
  • Remark 1.1
  • Definition 1.2: Constant maps to the critical point
  • Remark 1.3: Higher smoothness of Newton-Picard map
  • Lemma 2.1: Uniform bound
  • proof
  • Example 2.2: Constant maps to the critical point
  • Lemma 2.3
  • proof
  • Lemma 3.1: Complement
  • ...and 63 more