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-square 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-square 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?)

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?)

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 square norm 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 square norm of the $\mu$. to a critical point, 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?)

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-square 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-square 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?)