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

TL;DR

The paper establishes a general compactness theorem for gradient flow lines, including Morse and Floer cases, with a focus on exponential decay estimates for Floer cylinders.

Contribution

It introduces a unified compactness framework for gradient flow lines and proves a new exponential decay estimate for Floer cylinders.

Findings

Proved a general compactness result for gradient flow lines.

Established an exponential decay estimate for Floer cylinders.

Unified treatment of Morse and Floer gradient flow lines.

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.