Let $(M,\omega)$ be a Kahler manifold, and $E\rightarrow M$ be a holomorphic bundle. Suppose there is a hermitian-Einstein metric on $E$ with respect to $\omega$. That is a metric $H$ solving $$\Lambda_\omega F_H = \mu I.$$ Here $F_H\in End(E)$ is the curvature of the Chern connection, $\Lambda_\omega$ is the trace with respect to $\omega$, $\mu$ is a constant and $I$ is the identity endomorphism. My question is suppose $\omega'$ is another Kahler metric in the Kahler class of $\omega$, does there exists a metric $H'$ such that $$\Lambda_{\omega'}F_{H'} = \mu I?$$
This should be true since by the theorem of Donaldson-Uhlenbeck-Yau, existence of Hermitian-Einstein metrics is equivalent to the stability of $E$, and stability in turn depends only on the Kahler class, not on the Kahler metric itself. But is there a direct (preferably constructive) way to find $H'$ without having to appeal to this deep theorem? If $\omega' = \omega+\partial\bar\partial \phi$, I had hoped that $H' = e^{-\phi}H$ would work. But that does not seem to be the case.
Definition of stability - Recall that a holomorphic bundle $E\rightarrow M$ is called stable (with respect to the Kahler class $\alpha$) if for every proper coherent sub-sheaf $F$ such that $E/F$ is torsion free, we have $$\mu(F)<\mu(E).$$ Here the slope $\mu$ is defined by $$\mu(F) = \frac{c_1(F)\cdot \alpha^{n-1}}{\mathrm{rank}(F)}.$$