# Local behaviour of the gradient flow of an analytic function near the unstable set of a critical point

## TL;DR

This paper proves that for proper analytic functions on locally compact spaces, the gradient flow behaves well near critical points, extending Morse theory to certain singular spaces.

## Contribution

It shows that the first three conditions imply the fourth for locally compact spaces, and all conditions hold if the function is proper and analytic.

## Key findings

- Gradient flow is well-behaved near critical points under certain conditions.
- First three conditions imply the fourth condition in locally compact spaces.
- Proper analytic functions satisfy all five Morse-theoretic conditions.

## Abstract

This paper extends previous work from arxiv:1702.05223, which shows that the main theorem of Morse theory holds for a large class of functions on singular spaces, where the function and the underlying singular space are required to satisfy the five conditions explained in detail in the introduction to arxiv:1702.05223. The fourth of these conditions requires that the gradient flow of the function is well-behaved near the critical points, which is a very natural condition, but difficult to explicitly check for examples without a detailed knowledge of the flow. In this paper we prove a general result showing that the first three conditions always imply the fourth when the underlying space is locally compact. Moreover, if the function is proper and analytic then the first four conditions are all satisfied.