Refined analysis on finite-time singularities of almost-holomorphic harmonic map flow
Abstract
Building on the work of Song--Waldron~\cite{SongWaldron2024}, we establish stronger regularity and refined blow-up analysis results for almost-holomorphic harmonic map flow from a closed surface into a compact Kähler manifold with nonnegative holomorphic bisectional curvature. At each finite-time singularity, we improve the outer energy scale to O((T−t)/√(|ln(T−t)|)). We also prove that both the stress-energy tensor and the tension field converge strongly in L2 to those of the body map, by ruling out concentration of their L2 energy near the singular points. Consequently, the body map belongs to W2,2. Finally, an explicit gap theorem for the antiholomorphic energy of nonholomorphic harmonic spheres yields an explicit admissible threshold for the initial antiholomorphic energy under which these conclusions hold.
Related
- Refined Bohr Inequalities and Bohr Radii Involving Schwarz Mappings for Holomorphic Functions in Banach Spaces and JB*-Triples
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- Three-circle inequalities for almost harmonic functions on cylinders
- Refined sum formulas for finite multiple zeta values of level two
BibTeX
@misc{he2026refined,
title = {{Refined analysis on finite-time singularities of almost-holomorphic harmonic map flow}},
author = {Yihan He and Ze Li and Chong Song},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34132},
eprint = {2610.34132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34132},
}
RIS
TY - UNPB
TI - Refined analysis on finite-time singularities of almost-holomorphic harmonic map flow
AU - Yihan He
AU - Ze Li
AU - Chong Song
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34132
UR - https://arxiv.org/abs/2610.34132
AB - Building on the work of Song--Waldron~\cite{SongWaldron2024}, we establish stronger regularity and refined blow-up analysis results for almost-holomorphic harmonic map flow from a closed surface into a compact Kähler manifold with nonnegative holomorphic bisectional curvature. At each finite-time singularity, we improve the outer energy scale to O((T−t)/√(|ln(T−t)|)). We also prove that both the stress-energy tensor and the tension field converge strongly in L^2 to those of the body map, by ruling out concentration of their L^2 energy near the singular points. Consequently, the body map belongs to W^2,2. Finally, an explicit gap theorem for the antiholomorphic energy of nonholomorphic harmonic spheres yields an explicit admissible threshold for the initial antiholomorphic energy under which these conclusions hold.
ER -