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 $\Lambda_{\omega_g}\left(\sqrt{-1} R^{D^{h_0}}\right)$ of the Higgs connection is positive definite. Then for any Hermitian positive definite tensor $P\in \Gamma\left(M,E^*\otimes \bar E^*\right)$, there exists a unique smooth Hermitian metric $h$ on $E$ such that $$\Lambda_{\omega_g} \left(\sqrt{-1} R^{D^h}\right)=P.$$ We also establish quantitative Chern number inequalities for Higgs bundles.