Nonlinear evolution by mean curvature and isoperimetric inequalities
Nonlinear evolution by mean curvature and isoperimetric inequalities
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DOI：
10.4310/jdg/1211512640
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发表时间：
2006-06
影响因子：
2.5
通讯作者：
F. Schulze
中科院分区：
文献类型：
--
作者：
F. Schulze
在R^{n+1}中，以平均曲率的正幂k为法向速度的光滑紧致超曲面的演化，在k >= n-1时，改进了一定的“等周差”。由于奇点可能发展之前的体积为零，我们开发了一个弱水平集制定这样的流量，并表明上述单调性仍然是有效的。这证明了n <= 7的等周不等式。推广到具有非正截面曲率的完备单连通三维流形，给出了这类流形上欧氏等周不等式的一个新证明。
Evolving smooth, compact hypersurfaces in R^{n+1} with normal speed equal to a positive power k of the mean curvature improves a certain 'isoperimetric difference' for k >= n-1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n <= 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.