Iterative construction of Hermitian-Einstein metrics on stable bundles

Abstract: Let $E$ be a stable holomorphic vector bundle over a compact Kähler (or Gauduchon) manifold $(M,ωg)$. We show that for any real number $μ>0$ and any initial Hermitian metric $h_0$ on $E$, there exists a unique iteration sequence ${h_m}$ satisfying $$ Λ{ωg}\left(\sqrt{-1}R^{h{m+1}}\right) =(λE-μ)h{m+1}+μh_m, $$ and ${h_m}$ converges smoothly to a Hermitian-Einstein metric $h_\infty$ on $E$ satisfying $$ Λ{ω_g}\left(\sqrt{-1}R^{h{\infty}}\right) =λEh\infty, $$ where $λ_E\in \mathbb R$ 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.