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

In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles over compact complex manifolds. Let $(E,\theta )$ be a Higgs bundle over a compact Hermitian manifold $(M,{\omega }_g)$. Suppose that there exists a smooth Hermitian metric $h_0$ on $E$ such that the Hermitian-Yang-Mills tensor ${\mathrm{\Lambda }}_{{\omega }_g}\left(\sqrt{-1}{R}^{D^{h_0}}\right)$ of the Higgs connection 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 $${\mathrm{\Lambda }}_{{\omega }_g}\left(\sqrt{-1}{R}^{D^h}\right)=P\mathrm{.}$$ We also establish quantitative Chern number inequalities for Higgs bundles.