LOCAL BEHAVIOUR OF THE GRADIENT FLOW OF AN ANALYTIC FUNCTION NEAR THE UNSTABLE SET OF A CRITICAL POINT

In an earlier work (2019) we showed that the main theorem of Morse theory holds for a large class of functions on singular spaces, assumed to satisfy five conditions. The fourth of these conditions required 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 saying 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.