Stratification and Rectifiability of Harmonic Map Flows via Tangent Measures

In this paper, we investigate the stratification theory for ``suitable solutions" of harmonic map flows based on the spatial symmetry of tangent measures. Generally, suitable solutions are a category of solutions that satisfy both the localized energy inequality and the monotonicity formula. Building on the modifications and adjustments of quantitative stratifications and Reifenberg-rectifiable theory by Naber and Valtorta in the breakthrough research of harmonic maps (\emph{Ann. Math.} 185 (2017), 131-227), we confirm that each stratum in our model is rectifiable. Furthermore, for each time slice of the singular set, we establish the estimate of the Minkowski content and demonstrate its rectifiability, strengthening prior findings, which applied only to almost every time slice. Additionally, by making certain assumptions about the target manifolds to exclude specific tangent flows and measures, our analysis yields sharp improvements in the regularity of suitable solutions for harmonic map flows.