Partial regularity of heat flow of fractional harmonic maps to spheres

In this paper, we introduce the notation of stationary heat flow of $s$-harmonic maps from ${\mathbb{R}}^m$ into the unit sphere ${\mathbb{S}}^{\mathrm{\ell }-1}\hookrightarrow {\mathbb{R}}^{\mathrm{\ell }}$, for $0<s<1$, framed through the parabolic version of Caffarelli-Silvestre’s extension. We prove that any stationary heat flow of $s$-harmonic maps is smooth away from a closed set of vanishing $(m+2-2s)$-dimensional parabolic Hausdorff measure. This extends to the fractional settings the partial regularity theorem on the heat flow of harmonic maps into spheres by Chen-Li-Lin and Feldman.