科研之友
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摘要
The entropy is a natural geometric quantity which measures the complexity of a hypersurface in . It is non-increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow. In (Colding et al., J Differ Geom 95(1):53-69, 2013), Colding-Ilmanen-Minicozzi-White showed that within the class of closed smooth self-shrinking solutions of the mean curvature flow in , the entropy is uniquely minimized at the round sphere. They conjectured that, for , the round sphere minimizes the entropy among all closed hypersurfaces. Using an appropriate weak mean curvature flow, we prove their conjecture. For these dimensions, our approach also gives a new proof of the main result of Colding et al. (J Differ Geom 95(1):53-69, 2013) and extends its conclusions to compact singular self-shrinking solutions.
- 出版日期2016-12
