A sharp lower bound for the entropy of closed hypersurfaces up to dimension six

• 成果类型: Article
• 署名作者: Bernstein, Jacob; Wang, Lu
• 署名单位: Johns Hopkins University; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0659-3
• 发表日期: 2016
• 页码: 601-627
• 关键词: mean-curvature flow; plane-curves; level sets; SINGULARITIES; REGULARITY; shrinkers; motion

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.