The prescribed Hermitian-Yang-Mills flow I

In this paper, we introduce a broad class of flows, including the prescribed Hermitian-Yang-Mills flow: $$\frac{\mathrm{\partial }h}{\mathrm{\partial }t}=-{\mathrm{\Lambda }}_{{\omega }_g}\left({\sqrt{R}}^h\right)+P$$ where $P\in \mathrm{\Gamma }(M,E^{\ast }\otimes {\bar{E}}^{\ast })$ is a prescribed Hermitian tensor associated with a holomorphic vector bundle $E$ over a Kähler (or Hermitian) manifold $(M,{\omega }_g)$. We establish the long-time convergence of the flow to a limiting metric $h_{\mathrm{\infty }}$ and use it to solve the prescribed Hermitian-Yang-Mills tensor equation $${\mathrm{\Lambda }}_{{\omega }_g}\left({\sqrt{R}}^{h_{\mathrm{\infty }}}\right)=P,$$ for a general class of prescribed Hermitian tensors $P$. The crucial uniform $C^0$-estimate of $\{h(t)\}$ along the flow is obtained via a parabolic comparison principle.