Long-time behavior of the Hermitian-Yang-Mills flow on non-Kähler manifolds

In this paper, we study the long-time behavior of the Hermitian-Yang-Mills flow over compact Hermitian manifolds. We obtain the monotonicity of lower bound and upper bound of the eigenvalues of the mean curvature along the Hermitian-Yang-Mills flow. In the Gauduchon case, we show that the eigenvalues of the mean curvature converge to geometric invariants determined by the Harder-Narasimhan type. Furthermore, we generalize the Atiyah-Bott-Bando-Siu question to the non-Kähler case.