Abstract

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.

中文翻译：

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非局部几何演化的稳定性结果和分数平均曲率流的极限情况

摘要

我们为局部和非局部曲率引入了一致收敛的概念。然后，我们提出了一种抽象方法来证明相应几何流在水平集公式内的收敛性。我们应用这样一个一般理论来描述s -分数平均曲率流的极限为$秒\to 0^{+}$ 和 $秒\to 1^{-}.$与s -分数平均曲率流类似，我们引入了s -Riesz 曲率流的概念，并将其极限刻画为$秒\to 0^{-}.$ 最后，我们将极限行为讨论为 $r\to 0^{+}$r- Minkowski 内容的正则化产生的流。