RC-positivity, comparison theorems and prescribed Hermitian-Yang-Mills tensors I

In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem. Let $E$ be a holomorphic vector bundle over a compact K\"ahler manifold $(M,\omega_g)$. Suppose that there exists a smooth Hermitian metric $h_0$ on $E$ such that the Hermitian-Yang-Mills tensor $\Lambda_{\omega_g}\sqrt{-1} R^{h_0}$ is positive definite. Then for any Hermitian positive definite tensor $P\in \Gamma\left(M,E^*\otimes \overline E^*\right)$, there exists a unique smooth Hermitian metric $h$ on $E$ such that $$\Lambda_{\omega_g} \sqrt{-1} R^h=P.$$ The proof is based on a new comparison theorem for Hermitian-Yang-Mills tensors. Inspired by these results, we have also derived quantitative Chern number inequalities that apply to both holomorphic vector bundles and compact K\"ahler manifolds.