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The linearized Floer equation in a chart

Urs Frauenfelder, Joa Weber

TL;DR

The paper analyzes the Hessian of the area functional in a non-Darboux chart, introducing the concept of an almost extendable weak Hessian field to handle discontinuities. By decomposing the Hessian as $A=F+C$ into a continuous, Fredholm part and a lower-order perturbation, it applies Rabier's Fredholm framework to show the Robbin-Salamon operator ${\mathbb{D}}^u=\partial_s+A^u$ is Fredholm, with the restricted operator $\mathbb{D}^u_2=\partial_s+A^u_2$ sharing the same index. The main result proves the Fredholm property and index equality along connecting paths, even when the Hessian is not continuous, via a careful almost extendability construction and compact perturbation arguments. This advances Floer-theoretic analysis in non-smooth settings and supports extending Floer-type theory to Hamiltonian delay equations and related problems.

Abstract

In this article we are considering the Hessian of the area functional in a non-Darboux chart. This does not seem to have been considered before and leads to an interesting new mathematical structure which we introduce in this article and refer to as \emph{almost extendable weak Hessian~field}. Our main result is a Fredholm theorem for Robbin-Salamon operators associated to \emph{non-continuous} Hessians which we prove by taking advantage of this new structure.

The linearized Floer equation in a chart

TL;DR

The paper analyzes the Hessian of the area functional in a non-Darboux chart, introducing the concept of an almost extendable weak Hessian field to handle discontinuities. By decomposing the Hessian as into a continuous, Fredholm part and a lower-order perturbation, it applies Rabier's Fredholm framework to show the Robbin-Salamon operator is Fredholm, with the restricted operator sharing the same index. The main result proves the Fredholm property and index equality along connecting paths, even when the Hessian is not continuous, via a careful almost extendability construction and compact perturbation arguments. This advances Floer-theoretic analysis in non-smooth settings and supports extending Floer-type theory to Hamiltonian delay equations and related problems.

Abstract

In this article we are considering the Hessian of the area functional in a non-Darboux chart. This does not seem to have been considered before and leads to an interesting new mathematical structure which we introduce in this article and refer to as \emph{almost extendable weak Hessian~field}. Our main result is a Fredholm theorem for Robbin-Salamon operators associated to \emph{non-continuous} Hessians which we prove by taking advantage of this new structure.
Paper Structure (28 sections, 37 theorems, 209 equations, 3 figures)

This paper contains 28 sections, 37 theorems, 209 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The area functional in a Darboux chart
  3. Euclidean inner product and symplectic forms
  4. Associated anti-symmetric matrix
  5. Compatible almost complex structure
  6. Action functional
  7. First derivative and gradient
  8. Second derivative
  9. Para-Darboux Hessian
  10. Perturbed action functional
  11. Perturbed para-Darboux Hessian
  12. Fredholm operators
  13. Almost extendability
  14. Weak Hessian fields and decompositions
  15. Main theorem
  16. ...and 13 more sections

Key Result

Lemma 3.2

At any point $x\in \mathfrak{U}$ the linear map $B_x$ is invertible and satisfies for all $\xi,\eta\in{\mathbb{R}}^{2n}$.

Figures (3)

  • Figure 1: Basic path $\hat{u}$ and open cover of connecting path $u$ along time $[-T_-,T_+]$
  • Figure 2: Graphs of $\mathopen|\xi\mathclose|$ and $\xi$
  • Figure 3: Newton-Picard iteration for $f(r)=r^2-q$

Theorems & Definitions (108)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Definition 4.1
  • Lemma 4.2: Gradient
  • ...and 98 more