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 $(M,ω_g)$. 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 $$ Λ_{ω_g}\left(\sqrt{-1}R^{h_{m+1}}\right) =(λ_E-μ)h_{m+1}+μh_m, $$ and $\{h_m\}$ converges smoothly to a Hermitian-Einstein metric $h_\infty$ on $E$ satisfying $$ Λ_{ω_g}\left(\sqrt{-1}R^{h_{\infty}}\right) =λ_Eh_\infty, $$ 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.