A nonlocal curvature flow is investigated to evolve compact locally convex hypersurfaces in the Euclidean space \({\mathbb {E}}^{n+1}\). It is shown that the flow exists globally in all dimensions and deforms the evolving curve into an m-fold circle in the plane if \(n=1\) and drives the evolving hypersurface into a Euclidean sphere if \(n>1\).

中文翻译：

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演化的局部局部凸曲线和凸超曲面

研究了非局部曲率流，以在欧几里得空间\（{\ mathbb {E}} ^ {n + 1} \）中演化出紧凑的局部凸超曲面。结果表明，流动在所有维度全局存在和变形演化曲线成米倍圈在平面如果\（N = 1 \）和驱动器的演变超曲面成欧几里德球体如果\（N> 1 \）。