Continuation and convergence for the higher dimensional conformal Chern--Calabi flow
Abstract
We study the volume-normalized gradient flow of the squared Chern scalar curvature on compact Hermitian manifolds of complex dimension n≥3. The flow extends smoothly past a finite time if the spatial supremum of Chern scalar curvature belongs to L2 in time. A uniform curvature bound in the evolving Lp norm, for one fixed p>n, gives global existence and smooth convergence of the whole trajectory to a constrained critical metric. On Chern scalar-flat backgrounds, small initial Lp curvature gives exponential convergence to the background. A corresponding condition on the trace-free first Chern--Ricci form gives exponential convergence on Chern--Ricci-flat Hermitian and positive Kähler--Einstein backgrounds. The smallness thresholds are independent of the initial higher derivatives. We also construct constant Chern scalar curvature backgrounds of either sign with arbitrarily small smooth perturbations whose solutions, if global, cannot converge back to the background.
Related
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BibTeX
@misc{chen2026continuation,
title = {{Continuation and convergence for the higher dimensional conformal Chern--Calabi flow}},
author = {Hongyu Chen and Li Ma and Li Sheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31270},
eprint = {2610.31270},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31270},
}
RIS
TY - UNPB
TI - Continuation and convergence for the higher dimensional conformal Chern--Calabi flow
AU - Hongyu Chen
AU - Li Ma
AU - Li Sheng
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31270
UR - https://arxiv.org/abs/2610.31270
AB - We study the volume-normalized gradient flow of the squared Chern scalar curvature on compact Hermitian manifolds of complex dimension n≥3. The flow extends smoothly past a finite time if the spatial supremum of Chern scalar curvature belongs to L^2 in time. A uniform curvature bound in the evolving L^p norm, for one fixed p>n, gives global existence and smooth convergence of the whole trajectory to a constrained critical metric. On Chern scalar-flat backgrounds, small initial L^p curvature gives exponential convergence to the background. A corresponding condition on the trace-free first Chern--Ricci form gives exponential convergence on Chern--Ricci-flat Hermitian and positive Kähler--Einstein backgrounds. The smallness thresholds are independent of the initial higher derivatives. We also construct constant Chern scalar curvature backgrounds of either sign with arbitrarily small smooth perturbations whose solutions, if global, cannot converge back to the background.
ER -