 Gradient-flow dissipation identity

For the downward gradient flow $\dot{x}=-{\nabla }_{g}f$, the chain rule gives $df(x(t))/dt=-\Vert {\nabla }_{g}f(x(t)){\Vert }_g^2$. Integrating converts a bound on the decrease of $f$ into a bound on elapsed time wherever its Riemannian gradient is bounded away from zero.