Mean curvature nonpositivity and rational connectedness
Abstract
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 μω0,max(TX)=0 for some Gauduchon metric ω0, we show that the corresponding maximal destabilizing subsheaf of TX 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 Kähler manifold is not uniruled if and only if , for every Gauduchon metric ω, its tangent bundle admits a Hermitian metric with nonpositive ω-mean curvature. We also obtain an uniruledness criterion that includes the case where the maximal slope of TX is zero, and a Mumford-type characterization of rational connectedness. Under HN-nonnegativity of TX for one Gauduchon metric, rational connectedness, the vanishing of all holomorphic covariant tensors, and the condition κ+(X)=−∞ are equivalent.
Related
- Campana Rational Connectedness and Weak Approximation of Del Pezzo Orbifolds
- The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow
- The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture
- Singularities of Mean Curvature Flow with Bounded Total Curvature
- Existence of Lagrangian Mean Curvature Flow Singularities with Bounded Mean Curvature
BibTeX
@misc{zhang2026mean,
title = {{Mean curvature nonpositivity and rational connectedness}},
author = {Chuanjing Zhang and Xi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29268},
eprint = {2610.29268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29268},
}
RIS
TY - UNPB
TI - Mean curvature nonpositivity and rational connectedness
AU - Chuanjing Zhang
AU - Xi Zhang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29268
UR - https://arxiv.org/abs/2610.29268
AB - 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 Gauduchon metric ω_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 Kähler manifold is not uniruled if and only if , for every Gauduchon metric ω, its tangent bundle admits a Hermitian metric with nonpositive ω-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 κ^+(X)=−∞ are equivalent.
ER -