Exponential decay for sc-gradient flow lines
This paper is a contribution to the so-called ``polyfold theory'', mainly developed by \textit{H. Hofer}, \textit{K. Wysocki} and \textit{E. Zehnder} (see, e.g. [J. Eur. Math. Soc. (JEMS) 9, No. 4, 841--876 (2007; Zbl 1149.53053)]), whose goal is to provide a unified framework for the analytic study of moduli spaces occuring in various theories such as Morse theory, Gromov-Witten theory, Floer theory or symplectic field theory. The idea is to define a notion of ``sc-space'', a notion of convergence on these sc-spaces and a notion of linear maps between them. Out of that, one can define a new notion of smoothness and rewrite differential calculus in this settings. This gives rise to the analytical background on which polyfold theory is built. In Morse theory or Floer theory in particular, one considers an ``action'' function defined on some (possibly infinite-dimensional) Banach space and considers how its gradient trajectories are organized to produce some algebraic invariants. For polyfold theory, one needs to consider analogous functions defined on sc-spaces instead of Banach spaces. A key point in usual Morse and Floer theories is the fact that when a gradient trajectory converges to a non-degenerate critical point, then the convergence rate is exponential. In the paper under review, the authors prove that this exponential rate still holds in the sc-spaces framework.
scientific article; zbMATH DE number 6251858
scientific article