Mean curvature nonpositivity and rational connectedness

Abstract

Let $(X,\omega)$ 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 K\"ahler manifold $X$ satisfying $\mu_{\omega_0,\max}(T_X)=0$ for some Gauduchon metric $\omega_0$, we show that the corresponding maximal destabilizing subsheaf of $T_X$ splits off holomorphically and is polystable of slope zero with respect to every Gauduchon metric. Its complement, if nonzero, is HN-negative with respect to every such metric. In particular, a compact K\"ahler manifold is not uniruled if and only if , for every Gauduchon metric $\omega$, its tangent bundle admits a Hermitian metric with nonpositive $\omega$-mean curvature. We also obtain an uniruledness criterion that includes the case where the maximal slope of $T_X$ is zero, and a Mumford-type characterization of rational connectedness. Under HN-nonnegativity of $T_X$ for one Gauduchon metric, rational connectedness, the vanishing of all holomorphic covariant tensors, and the condition $\kappa^+(X)=-\infty$ are equivalent.