# Symplectic stability on manifolds with cylindrical ends

## TL;DR

This paper extends symplectic stability results to certain noncompact manifolds with cylindrical ends, relaxing previous convexity and completeness conditions, and demonstrating the stability under specific growth conditions.

## Contribution

It introduces a new symplectic stability theorem for noncompact manifolds with cylindrical ends, removing convexity and completeness assumptions.

## Key findings

- Symplectic stability holds under natural growth conditions.
- Explicit examples demonstrate applicability.
- Retains end structure while relaxing conditions.

## Abstract

A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact symplectic manifolds. The notion of Eliashberg-Gromov convex ends provides a natural restricted setting for the study of analogs of Moser's symplectic stability result in the noncompact case, and this has been significantly developed in work of Cieliebak-Eliashberg. Retaining the end structure on the underlying smooth manifold, but dropping the convexity and completeness assumptions on the symplectic forms at infinity we show that symplectic stability holds under a natural growth condition on the path of symplectic forms. The result can be straightforwardly applied as we show through explicit examples.