nLab
gradient flow

Idea

For $(X,g)$ a Riemannian manifold and $f:X\to ℝ$ a smooth function, let

$$\nabla f:=g^{-1}(d_{\mathrm{dR}}f)\in \Gamma (TX)$$\nabla f := g^{-1}(d_{dR} f) \in \Gamma(T X)

be the gradient vector field of $X$. The flow induced by this on $X$ is the gradient flow of $f$.

Examples

• 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.)