For $(X,g)$ a Riemannian manifold and $f : X \to \mathbb{R}$ a smooth function, let
be the gradient vector field of $X$. The flow induced by this on $X$ is the gradient flow of $f$.
Yang-Mills instantons are the gradient flow trajectories of the Chern-Simons action functional.
Ricci flow is the gradient flow of the action functional of dilaton gravity.
(This is a key part of Perelman’s proof of the Poincare conjecture.)