The prescribed Hermitian-Yang-Mills flow II

Abstract

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 Kähler manifold $(M,\ddot{I}{‰}_g)$. Suppose that for every proper coherent subsheaf $F\subset E$, the following inequality holds: $$de{g}_{\ddot{I}{‰}_g}(F)<de{g}_{\ddot{I}{‰}_g}(E).$$ Then, for any initial Hermitian metric $h_0$ on $E$ and any positive-definite Hermitian tensor $P\in \hat{I}“(M,E^{\ast }\otimes {\overline{E}}^{\ast })$, the prescribed Hermitian-Yang-Mills flow $$\frac{\partial h}{\partial t}=-\hat{I}{›}_{\ddot{I}{‰}_g}\left(\sqrt{-1}{R}^h\right)+P,$$ admits a global smooth solution on $[0,\infty )$. Moreover, as $t\to \infty$, the flow converges smoothly to a Hermitian metric $h_{\infty }$ on $E$ satisfying $$\hat{I}{›}_{\ddot{I}{‰}_g}\left(\sqrt{-1}{R}^{h_{\infty }}\right)=P.$$ As an application, we establish that on a Fano manifold $M$, for any Hermitian metric form $\ddot{I}‰$ and any positive-definite Hermitian tensor $P\in \hat{I}“(M,T^{\ast 1,0}M\otimes T^{\ast 0,1}M)$, there exists a unique Hermitian metric tensor $h$ on $T^{1,0}M$ such that $$\hat{I}{›}_{\ddot{I}}‰\left({\sqrt{R}}^h\right)=P.$$ This may be viewed as an analogue of the Calabi-Yau theorem for Fano manifolds.