Exponential decay for sc-gradient flow lines

TL;DR

This paper introduces sc-action functionals and proves that their gradient flow lines exhibit uniform exponential decay under Morse conditions, using interpolation inequalities instead of Sobolev inequalities for the analysis.

Contribution

It establishes exponential decay estimates for sc-gradient flow lines using interpolation inequalities, advancing the understanding of sc-structures in Floer theory.

Findings

Proves uniform exponential decay of sc-gradient flow lines under Morse conditions.

Develops decay estimates using interpolation inequalities, independent of source space dimension.

Lays groundwork for constructing M-polyfold bundles related to broken flow lines.

Abstract

In this paper we introduce the notion of sc-action functionals and their sc-gradient flow lines. Our approach is inspired by Floer's unregularized gradient flow. The main result of this paper is that under a Morse condition sc-gradient flow lines have uniform exponential decay towards critical points. The ultimate goal for the future is to construct a M-polyfold bundle over a M-polyfold such that the space of broken sc-gradient flow lines is the zero set of a appropriate sc-section. Here uniform exponential decay is essential. Of independent interest is that we derive exponential decay estimates using interpolation inequalities as opposed to Sobolev inequalities. An advantage is that interpolation inequalities are independent of the dimension of the source space.