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.

Building on the quantitative stratifications and Reifenberg-rectifiable theory developed by Naber and Valtorta in breakthrough research of harmonic maps (\emph{Ann.

Math.} 185 (2017), 131-227), we prove that each time slice of the singular set in our model is rectifiable.

By making some additional assumptions about the target manifolds to exclude specific tangent flows and measures, we can also obtain a sharp regularity of suitable solutions for harmonic map flows.