Almost complex structure invariant by hamiltonian action I was reading J.Evans lectures, in particular: Not necesary to read to understand question.
At some point, he claims that given a compact symplectic manifold $(M,\omega)$ and a hamiltonian action
from $\mathbb{S}^1$
on $M$. Then it is possible to construct an invariant almost complex structure $J\in End(M)$.
The way he does it is as follows: Let $g$ be an $\mathbb{S}^1$ invariant metric on $M$
which can be obtained by averaging over the metrics on $g$. (Details are not given though).
Then, defines $A$ such as the unique matrix so that:
$$
\omega(\cdot,\cdot)=g(A\cdot,\cdot)
$$
Then we get an almost complex structure from $A$ in the "usual" way:
$$
J=\sqrt{AA^T}A
$$
Where everything is well-defined because $A$ is skew-symmetric
and therefore $AA^T$ is symmetric and positive with respect to $g$.
The final claim is that, since $J$ is unique it must be almost invariant.
However, $J$ is not unique as an almost complex structure compatible with $\omega$,
there usually many of those. So, what does he mean by unique?
 A: First off, the averaging over metrics is accomplished by taking any metric $\widetilde{g}$, and defining a new metric $g$ by
$$
g_p:= \int_{S^1}(\Phi^*_\theta \widetilde{g})_p \,d\theta,
$$
where $\Phi_\theta:M\to M$ is the $S^1$ action on $M$, and $\Phi^*_\theta\widetilde{g}$ is the pullback under this action, defined by
$$
(\Phi^*_\theta\widetilde{g})_p(u_p,v_p) := \widetilde{g}_{\Phi_\theta(p)}(T_p\Phi_\theta(u_p), T_p\Phi_\theta(v_p)).
$$
It seems a bit strange to me to phrase this in terms of `convexity', though, as Evans does.
Secondly, by `uniqueness' it seems Evans means uniqueness of the map $A$, rather than of an almost complex structure. By this, I interpret that he means since $\omega(\cdot, \cdot) = g(A\cdot,\cdot)$, and $\omega$ and $g$ are both $S^1$-invariant, so too is $A$ (which then implies $J = \sqrt{AA^T}^{-1}A$ is invariant). It's really the non-degeneracy of $g$ that is being invoked, so again it's a bit strange to phrase this in terms of uniqueness of $A$. I guess the idea is that if $A'$ denotes the map $A$ transformed via $\Phi_\theta$ for some $\theta\in S^1$, then invariance of $\omega$ and $g$ implies that $\omega(\cdot,\cdot)=g(A'\cdot,\cdot)$, which then implies that $A=A'$, i.e. $A$ is $S^1$-invariant.
