Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$.
Assume that $\omega$ is an arbitrary symplectic structure on $U$.


Is there a Riemannian metric on $U$ with the following property?



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> For every smooth function $f: U \to \mathbb{R}$, we have $J \nabla f = X\_f$ where $X\_f$ is the Hamiltonian vector field corresponding to $f$
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> 
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A relation between gradient vector field and Hamiltonian vector field
You are requiring that 


$$g(\nabla f, X) = \omega (J\nabla f, X)$$


for all $f$ and for all $X$. So if $g$ exists, $g$ is of the form


$$g(X, Y) = \omega (JX, Y).$$


But the right hand side is not positive definite in general (for example $\omega = -\sum dx^i \wedge dy^i$). 


