The Hermitian metric $H$ along the fibers of the holomorphic vector bundle $E$ over a K\"ahler manifold $(M,\omega)$ is Hermitian-Einstein if $$\sqrt[]{-1}\Lambda_\omega F_H =
\gamma I$$ at every point of $M$, where $\gamma$ is a global constant and $I$ is the identity
endomorphism of $E$ and $F_H$ is the Chern curvature form of $(E, h)$. Here $\Lambda_\omega$ is the contraction with $\omega$ which means 
$$F_H\wedge\omega^{n-1}=(n-1)!\Lambda_\omega F_H\frac{\omega^n}{n!}$$

If a vector bundle $E$ admit Hermitian-Einstein metric, then the trace free part of $F_H^o=F_H-(\frac{1}{rk E}Tr F_H)$ is harmonic form.

Now take the integral 
$$D_\lambda(H_0,H)=\int_V(R_2(H,H_0)-2\lambda R_1(H,H_0)\wedge \omega)\wedge \frac{\omega^{m-1}}{m!}$$
where $\bar\partial\partial R_2(H,H_0)=4\pi^2(\pi_1(E,H)-\pi_1(E,H_0))$ and $\bar\partial\partial R_1(H_1,H_0)=2\pi(c_1(E,H)-c_1(E,H_0))$ where $\pi_1(E,H_0)$ is the first Pontryagin form.

Now if you take the Donaldson flow (which is like Kahler-Ricci flow version(due to Yau) for  Vector bundles) . Take $H_t=H_0.h_t$
$$\frac{\partial h_t}{\partial t}=-2h_t(\Lambda F_{h_t}-\lambda Id)$$

Now if we choose the Initial metric as smooth Hermitian metric and such flow admit a solution $H_\infty$ called Hermitian-Einstein metric then for all other Hermitian metric $H$ we have $D_\lambda(H,H_\infty)>0$, this tells us that such flow admit at most one solution and we get unicity of the solution of Hermitian-Einstein metric. But for singular Hermitian-Einstein metric in general we don't have $D_\lambda(H,H_\infty)>0$ and it can be negative. So we need to add some condition on the singular Hermitian metric to run the flow and get uniqueness. Something like vanishing Lelong number(I don't know it works).



 As a motivation when we consider parabolic stability for framed Vector bundle on pair $(X,D)$ we need to add some condition on initial metric such that to control blowing up along divisor and get uniqueness of the solutions . A lot of papers published without considering this important point. As an example Existence of Hermitian-Einstein metric on stable Orbifold bundle on orbifolds which initial metric can be non-smooth in general