On gradient flows initialized near maxima

TL;DR

This paper demonstrates that on a generic smooth function and metric on a closed Riemannian manifold, gradient flows near maxima tend to converge to specific minima, revealing a structured max-min relation.

Contribution

It establishes a generic property of gradient flows near maxima, showing convergence to particular minima and introduces the max-min graph to encode these relations.

Findings

Gradient flows near maxima typically converge to two specific minima.

The results hold for generic functions and metrics on closed Riemannian manifolds.

A max-min graph encodes the relations between maxima and minima.

Abstract

Let $(M,g)$ be a closed Riemannian manifold, and let $F:M \to \mathbb{R}$ be a smooth function on $M$. We show the following holds generically for the function $F$: for each maximum $p$ of $F$, there exist two minima, denoted by $m_+(p)$ and $m_-(p)$, so that the gradient flow initialized at a random point close to $p$ converges to either $m_-(p)$ or $m_+(p)$ with high probability. The statement also holds for $F \in C^\infty(M)$ fixed and a generic metric $g$ on $M$. We conclude by associating to a given a generic pair $(F, g)$ what we call its max-min graph, which captures the relation between minima and maxima derived in the main result.