The following question is related to Singular Yang-Mills theory

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 $$D_\lambda(H_0,H)=\int_M(R_2(H,H_0)-2\lambda R_1(H,H_0)\wedge \omega)\wedge \frac{\omega^{n-1}}{n!}$$ 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, lets define the Donaldson flow (which is like Kahler-Ricci flow version(due to Yau) for Vector bundles) .

Let $\mathcal M$ be the Frechet space of hermitian metrics on $E$, we define $\theta$ as follows $\mathcal X\in T_H\mathcal M$, $$\mathcal X(u,v)=H(\theta(\mathcal X)u,v)$$

Now Take $H_t=H_0.h_t$ then $$\frac{\partial H_t}{\partial t}=-2\theta^{-1}(\Lambda F_{H_t}-\lambda Id)$$

can be written as follows

$$\frac{\partial h_t}{\partial t}=-2h_t(\Lambda F_{h_t}-\lambda Id)$$

called Yau-Donaldson flow

Hence we are facing with a nonlinear PDE strictly parabolic since we can write it as follows

$$\frac{\partial h_t}{\partial t}=-\Delta h_t-\{\Lambda_{H_0}-\lambda Id,h\}+2i\Lambda\bar\partial h_t.h_t^{-1}\wedge \partial_0 h_t$$

where $\{,\}$ represent anti-commutator of $End(E)$. In fact the existence of the solutions of such flow correspond to local inverse theorem of Banach in smooth setting, but for singular setting, I don't know(since perturbation method does not work).

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 initial singular Hermitian metric to run the Yau-Donaldson flow and get uniqueness. Something like vanishing Lelong number. Which information on the initial metric is needed to get the unique solution for Yau-Donaldson flow?(Here I mean well defined singular Hermitian metric, since in general such metric is not well defined)

As a motivation, when we consider parabolic stability for framed Vector bundle on pair with a snc divisor $(X,D)$ we need to add some condition on initial metric such that to control blowing up along divisor and get uniqueness of the solutions . Recently a lot of papers published like mushroom 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. Or finding twisted Hermitian-Einstein metric on holomorphic fiber space needed to assume initial metric to be singular, see my question in MO

Or from the work of Y.T. Siu for existence of Hermitian-Einstein metric on reflexive sheaf

Let $(X,\omega)$ be a compact Kahler manifold and $\mathcal E$ be a reflexive sheaf on $X$. Then there exists a resolution $0\to \mathcal E\to E_0\to E_1\to E_2\to\cdots$ of vector bundles $E_i$ and if we take a hermitian metric on vector bundle $E_0$ then it induces a singular hermitian metric which is well defined on $\mathcal E$ on $X$ outside of closed subset $S$ of codim 4. and then they start a heat equation with such initial hermitian metric which is not smooth in general and they didn't proof the uniqueness of solutions and took it as trivial fact