Stable minimal hypersurfaces in RN+1+l with singular set an arbitrary closed Kc{0}xRl

• 成果类型: Article
• 署名作者: Simon, Leon
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2023.197.3.4
• 发表日期: 2023
• 页码: 1205-1234
• 关键词: strong maximum principle

摘要

With respect to a C infinity metric which is close to the standard Euclidean metric on RN+1+e, where N > 7 and $ > 1 are given, we construct a class of embedded (N + )-dimensional pound hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset K C {0} x Re. Thus the question is settled, with a strong affirmative, as to whether there can be gaps or even fractional dimensional parts in the singular set. Such questions, for both stable and unstable minimal submanifolds, remain open in all dimensions in the case of real analytic metrics and, in particular, for the standard Euclidean metric.The construction used here involves the analysis of solutions u of the symmetric minimal surface equation on domains 52 C Rn whose symmetric graphs (i.e., {(x, xi) E 52 x Rm : |xi| = u(x)}) lie on one side of a cylindrical minimal cone including, in particular, a Liouville type theorem for complete solutions (i.e., the case 52 = Rn).