In Donaldson's paper Symplectic submanifolds and almost-complex geometry, he mentioned that for each point $p$ in a compact almost-Kähler manifold $(V,\omega ,J)$, there exists a Darboux chart $\varphi_p\colon B^{2n} (1)\to V$ such that $\varphi_p (0)=p$ and the derivative of $\varphi_p$, measured with respect to the Levi-Civita connection and the Riemannian metric induced by $J$ and $\omega$, is bounded and the upper-bound is independent of the choice of the point $p$. Why is this the case? I don't know how to attack this.
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$\begingroup$ Just a random thought - I haven't thought about symplectic geometry in a while: is it possible that charts on balls with fixed radius $\rho > 0$ are allowed? Otherwise it would seem a bit weird given that it could presumably be that $\mathrm{vol}(V) < \omega_{2n}$. If a radius $\rho \in (0,1)$ were admissible, then it seems you could do a covering argument and translate your charts, no? $\endgroup$– Leo MoosApr 9, 2021 at 17:08
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