Exponential convergence of a Morse gradient flow
= Exponential convergence of a Morse gradient flow
{title2=$d(x(t),z)=O(e^{-2t})$}
The <Morse lemma> gives coordinates $f=f(z)-|u|^2+|v|^2$. Choose a <Riemannian metric> Euclidean in these coordinates. A downward <gradient flow> converging to $z$ has $u=0$ eventually, while $v(t)=e^{-2(t-t_0)}v(t_0)$. Thus its <Riemannian distance> from $z$ decays exponentially. A <partition of unity> makes this a global metric choice; changing the metric may change the trajectories.