Existence of twisted Hermitian-Einstein metrics on unstable vector bundles

In this paper, we demonstrate that twisted Hermitian-Einstein metrics on holomorphic vector bundles exist without obstruction. More precisely, for an arbitrary holomorphic vector bundle $E$ over a compact Kähler manifold $(M,{\omega }_g)$, we prove that the twisted Hermitian-Einstein equation $${\mathrm{\Lambda }}_{{\omega }_g}\left(\sqrt{-1}{R}^h\right)=\lambda h+P$$ admits a unique smooth solution $h$, provided that $P\in \mathrm{\Gamma }(M,E^{\ast }\otimes {\bar{E}}^{\ast })$ is positive-definite and $\lambda <{\lambda }_E^{-}$. The constant ${\lambda }_E^{-}$ is intrinsically associated with the stability constant of $E$. This result extends the classical Donaldson-Uhlenbeck-Yau (DUY) theorem for stable bundles and, in the limit $P\to 0$, gives a new proof of the DUY theorem. As an application, we obtain an intrinsic Chern number inequality for unstable vector bundles: $${\int }_M((r-1)c_1(E{)}^2-2r{c}_2(E))\wedge {\omega }_g^{n-2}\le \lfloor \frac{r^2}{4}\rfloor \frac{({\lambda }_E^{+}-{\lambda }_E^{-}{)}^2}{4{\pi }^2{n}^2}{\int }_M{\omega }_g^n\mathrm{.}$$