Iterative construction of Hermitian-Einstein metrics on stable bundles

Let $E$ be a stable holomorphic vector bundle over a compact Kähler (or Gauduchon) manifold $(M,{\omega }_g)$. We show that for any real number $\mu >0$ and any initial Hermitian metric $h_0$ on $E$, there exists a unique iteration sequence $\{h_m\}$ satisfying $${\mathrm{\Lambda }}_{{\omega }_g}\left(\sqrt{-1}{R}^{h_{m+1}}\right)=({\lambda }_E-\mu )h_{m+1}+\mu h_m,$$ and $\{h_m\}$ converges smoothly to a Hermitian-Einstein metric $h_{\mathrm{\infty }}$ on $E$ satisfying $${\mathrm{\Lambda }}_{{\omega }_g}\left(\sqrt{-1}{R}^{h_{\mathrm{\infty }}}\right)={\lambda }_E{h}_{\mathrm{\infty }},$$ where ${\lambda }_E\in \mathbb{R}$ is the stability constant. A key feature of this proof is that it is independent of Donaldson’s variational framework and applies to non-Kähler manifolds.