The prescribed Hermitian-Yang-Mills flow II

We prove an analogue of the classical Donaldson-Uhlenbeck-Yau theorem by using the prescribed Hermitian-Yang-Mills flow. Let $E$ be a holomorphic vector bundle over a compact K\"ahler manifold $(M,\omega_g)$. Suppose that for every proper coherent subsheaf $F\subset E$, the following inequality holds: $$ deg_{\omega_g}(F)<deg_{\omega_g}(E). $$ Then, for any initial Hermitian metric $h_0$ on $E$ and any positive-definite Hermitian tensor $P\in \Gamma(M,E^*\otimes \overline E^*)$, the prescribed Hermitian-Yang-Mills flow $$ \ \frac{\partial h}{\partial t} = -\Lambda_{\omega_g}\left(\sqrt{-1}\, R^h\right) + P, $$ admits a global smooth solution on $[0,\infty)$. Moreover, as $t\rightarrow\infty$, the flow converges smoothly to a Hermitian metric $h_\infty$ on $E$ satisfying $$ \Lambda_{\omega_g}\left(\sqrt{-1}\, R^{h_\infty}\right) = P. $$ As an application, we establish that on a Fano manifold $M$, for any Hermitian metric form $\omega$ and any positive-definite Hermitian tensor $P\in\Gamma(M,T^{*1,0}M\otimes T^{*0,1}M)$, there exists a unique Hermitian metric tensor $h$ on $T^{1,0}M$ such that $$ \Lambda_\omega\left(\sqrt R^h\right)=P.$$ This may be viewed as an analogue of the Calabi-Yau theorem for Fano manifolds.