Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. Let $<.,>$ denote a $G$-invariant inner product on $\mathfrak{g}$.


Let $(M,\omega)$ be a symplectic compact manifold endowed with a hamiltonian action of $G$, and let $\mu : M \longrightarrow \mathfrak{g}^\*,$ be a moment map associated to this action. We fix a Riemannian metric $g$ on $M$.


I have read somewhere that every gradient flow line $\phi\_x(t)$ of the norm-squared of the moment map $\mu$ begins and ends at a critical point, i.e $\lim\_{t \rightarrow + \infty} \phi\_t(x) $ and $\lim\_{t \rightarrow - \infty} \phi\_t(x) $ exist, and they are both critical points of the norm-squared of $\mu$. but I couldn't find a proof of this fact anywhere, Hopefully someone can help.


(This is a follow-up question to This one: [The negative gradient flow of a Morse-Bott function on a compact manifold converges to a critical point?](https://mathoverflow.net/questions/425477/the-negative-gradient-flow-of-a-morse-bott-function-on-a-compact-manifold-conver/429039#429039))



The norm-squared of a moment map behaves like a Morse-Bott function
As requested I am submitting my comment as an answer.
The desired statement is proved in [Gradient flow of the norm squared of a moment map](https://www.e-periodica.ch/cntmng?pid=ens-001%3A2005%3A51%3A%3A14) by Eugene Lerman who attributes the proof to Duistermaat.


