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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?)

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  • $\begingroup$ For a Hamiltonian $S^1$-action with Hamiltonian $H$, the square norm of the Hamiltonian may not be differentiable on the set $\{H=0\}$. For example when $\{H=0\}$ is disjoint from the fixed point set, then the square norm is not differentiable anywhere along it. So please clarify what you mean by critical points. $\endgroup$
    – Nick L
    Commented Aug 25, 2022 at 11:46
  • $\begingroup$ Also it may help if you could link to the place where you read it (if possible). $\endgroup$
    – Nick L
    Commented Aug 25, 2022 at 11:53
  • $\begingroup$ @NickL, the definition of critical points that I'm using is the following: $\textbf{Def}$: we say that $x \in M$ is a critical point of $\vert \vert \mu \vert \vert ^2 $ if $d_x \vert \vert \mu \vert \vert ^2 =0$. $\endgroup$
    – Mira
    Commented Aug 26, 2022 at 0:34
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    $\begingroup$ Ah ok, that is usually referred to as the norm squared. Square norm usually means $||\mu||$. Indeed the norm squared is differentiable. $\endgroup$
    – Nick L
    Commented Aug 26, 2022 at 10:45
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    $\begingroup$ This is proved in Gradient flow of the norm squared of a moment map by Eugene Lerman who attributes the proof to Duistermaat. $\endgroup$
    – Gustavo Granja
    Commented Aug 28, 2022 at 20:03

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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 by Eugene Lerman who attributes the proof to Duistermaat.

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