Let E be a holomorphic bundle over algebra surface X, let $H$ be a Hermitian metric of $E$, recall the Hermitian-Yang-mills equation is $\wedge F_H=\lambda.1$.

Let $H_t$ be Hermitian metrics over $E$ parametrized by $t$, Donaldson in [1] consider the following flow equation: \begin{equation} H_t^{-1}\frac{\partial H_t}{\partial t}=-2i(\wedge F_{H_t}-\lambda.1),\;\;H_t|_{t=0}=H_0, \label{flow} \end{equation} for some initial metric $H_0$.

In [1], page 13, there is a note: If $E$ is indecomposable and has a solution $K$ to the Hermitian-Yang-mills equation, then for any initial condition $H_0$, the corresponding solution $H_t$ of the flow equation converges in $\mathcal{C}^{\infty}$ to $K$ as $t\to\infty$. In addition, consider the distance function $\sigma$ between two metric $H_t,K$, $\sigma(H_t,K):=Tr(H_t^{-1}K)+Tr(K^{-1}H_t)-2\;\mathrm{rank}\;E$, then we have a bound \begin{equation} \|(\frac{\partial}{\partial t}+\Delta)\sigma(K,H_t)\|_{L^1}\leq -const.\|\sigma(K,H_t)\|_{L^1} \end{equation} and $\sigma$ decays exponentially.

My question is how to verify these two claims:

(A)If a solution $K$ exists, then the flow convergence to the solution in $\mathcal{C}^{\infty}$.

(B)This convergence is exponentially decays.

[1] S.Donaldson, Anti Self-Dual Yang-Mills Connections over Complex Algebraic Surfaces and Stable Vector Bundles

Let E be a holomorphic bundle over algebra surface X, let $H$ be a Hermitian metric of $E$, recall the Hermitian-Yang-mills equation is $\wedge F_H=\lambda.1$.

Let $H_t$ be Hermitian metrics over $E$ parametrized by $t$, Donaldson in [1] consider the following flow equation: \begin{equation} H_t^{-1}\frac{\partial H_t}{\partial t}=-2i(\wedge F_{H_t}-\lambda.1),\;\;H_t|_{t=0}=H_0, \label{flow} \end{equation} for some initial metric $H_0$.

In [1], page 15, there is a note: If $E$ is indecomposable and has a solution $K$ to the Hermitian-Yang-mills equation, then for any initial condition $H_0$, the corresponding solution $H_t$ of the flow equation converges in $\mathcal{C}^{\infty}$ to $K$ as $t\to\infty$. In addition, consider the distance function $\sigma$ between two metric $H_t,K$, $\sigma(H_t,K):=Tr(H_t^{-1}K)+Tr(K^{-1}H_t)-2\;\mathrm{rank}\;E$, then we have a bound \begin{equation} \|(\frac{\partial}{\partial t}+\Delta)\sigma(K,H_t)\|_{L^1}\leq -const.\|\sigma(K,H_t)\|_{L^1} \end{equation} and $\sigma$ decays exponentially.

My question is how to verify these two claims:

(A)If a solution $K$ exists, then the flow convergence to the solution in $\mathcal{C}^{\infty}$.

(B)This convergence is exponentially decays.

[1] S.Donaldson, Anti Self-Dual Yang-Mills Connections over Complex Algebraic Surfaces and Stable Vector Bundles