The nonlocal mean curvature flow of periodic graphs

We establish the well-posedness of the nonlocal mean curvature flow of order $\alpha \in (0,1)$ for periodic graphs on ${\mathbb{R}}^n$ in all subcritical little Hölder spaces ${\mathrm{h}}^{1+\beta }({\mathbb{T}}^n)$ with $\beta \in (0,1)$. Furthermore, we prove that if the solution is initially sufficiently close to its integral mean in ${\mathrm{h}}^{1+\beta }({\mathbb{T}}^n)$, then it exists globally in time and converges exponentially fast towards a constant. The proofs rely on the reformulation of the equation as a quasilinear evolution problem, which is shown to be of parabolic type by a direct localization approach, and on abstract parabolic theories for such problems.