Existence of infinitely many minimal hypersurfaces in positive Ricci curvature

• 成果类型: Article
• 署名作者: Marques, Fernando C.; Neves, Andre
• 署名单位: Princeton University; Imperial College London
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0716-6
• 发表日期: 2017
• 页码: 577-616
• 关键词: regularity; THEOREM; SPACE

摘要

In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.