Mean curvature nonpositivity and rational connectedness

Let $(X,ω)$ be a compact Gauduchon manifold and let $E$ be a holomorphic vector bundle on $X$. We establish equivalent conditions for the existence of a Hermitian metric on $E$ with nonpositive mean curvature. For a non-uniruled compact Kähler manifold $X$ satisfying $μ_{ω_0,\max}(T_X)=0$ for some G