Inserted: 4 mar 2020
Last Updated: 28 dec 2020
Journal: Comm. Partial Differential Equations
We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of $s$-fractional mean curvature flows as $s\to 0^+$ and $s\to 1^-$. In analogy with the $s$-fractional mean curvature flows, we introduce the notion of $s$-Riesz curvature flows and characterize its limit as $s\to 0^-$. Eventually, we discuss the limit behavior as $r\to 0^+$ of the flow generated by a regularization of the $r$-Minkowski content.