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 $O((T-t)/\sqrt{|\ln(T-t)|})$. 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.