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 Kähler manifold $(M,{\omega }_g)$. Suppose that there exists a smooth Hermitian metric $h_0$ on $E$ such that the Hermitian-Yang-Mills tensor ${\mathrm{\Lambda }}_{{\omega }_g}\sqrt{-1}{R}^{h_0}$ is positive definite. Then for any Hermitian positive definite tensor $P\in \mathrm{\Gamma }(M,E^{\ast }\otimes {\bar{E}}^{\ast })$, there exists a unique smooth Hermitian metric $h$ on $E$ such that $${\mathrm{\Lambda }}_{{\omega }_g}\sqrt{-1}{R}^h=P\mathrm{.}$$ 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 Kähler manifolds.