Extensions of Schoen-Simon-Yau and Schoen-Simon theorems via iteration à la De Giorgi

• 成果类型: Article
• 署名作者: Bellettini, Costante
• 署名单位: University of London; University College London
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01317-0
• 发表日期: 2025
• 页码: 1-34
• 关键词: minimal-surfaces; hypersurfaces; REGULARITY; CURVATURE

摘要

We give an alternative proof of the Schoen-Simon-Yau curvature estimates and associated Bernstein-type theorems (Schoen et al. in Acta Math. 134:275-288, 1975), and extend the original result by including the case of 6-dimensional (stable minimal) immersions. The key step is an epsilon-regularity theorem, that assumes smallness of the scale-invariant L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L<^>{2}$\end{document} norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension n >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq 2$\end{document}, that may have a singular set Sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Sigma $\end{document} of locally finite Hn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}<^>{n-2}$\end{document}-measure, and that is weakly close to a hyperplane. (In fact, if the Hn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}<^>{n-2}$\end{document}-measure of the singular set vanishes, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an epsilon-regularity theorem, that assumes smallness of the scale-invariant L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L<^>{2}$\end{document} tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen-Simon theorem (Schoen and Simon 34:741-797, 1981). In both epsilon-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method & agrave; la De Giorgi (from the linear De Giorgi-Nash-Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both epsilon-regularity theorems the method gives explicit constants that quantify the required smallness.)