发表机构
Beijing Institute of Mathematical Sciences and Applications; Mathematisches Institut Universität Freiburg（北京国际数学研究中心; 弗赖堡大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究建立了分支稳定极小浸入超曲面非分支奇异集的Hausdorff维数精确界，构造了达到该界的例子，并以广义Schoen不等式和分支分层定理为核心证明工具，推进了极小曲面正则性理论的研究。
AI 中文摘要
我们针对奇异集具有局部有限$\u27a4^{n-2}$测度的分支稳定极小浸入超曲面，建立了其非分支奇异集的Hausdorff维数的精确界：当$n=2$时非分支奇异集为空，当$n=3$时为离散集，当$n\geq4$时其Hausdorff维数至多为$n-3$。我们还在$\mathbb{R}^4$中构造了一个非平坦的稳定极小锥，它源自一个分支极小浸入，其顶点是一个非分支奇点。通过与欧氏因子作乘积，我们得到了非分支奇异集的Hausdorff维数恰好为$n-3$的例子，表明我们的正则性界在每个维数$n\geq3$下都是精确的。我们证明的主要要素是一个广义的Schoen不等式，以及对应的在平稳经典锥和超平面并集附近的分支分层定理。
英文摘要
We establish a sharp bound on the Hausdorff dimension of the non-branch singular set of branched stable minimal immersed hypersurfaces whose singular sets have locally finite $\mathcal H^{n-2}$-measure: the non-branch singular set is empty when $n=2$, discrete when $n=3$, and has Hausdorff dimension at most $n-3$ when $n\geq4$. We also construct a non-flat stable minimal cone in $\mathbb R^4$ arising from a branched minimal immersion whose vertex is a non-branch singularity. Taking products with Euclidean factors yields examples whose non-branch singular sets have Hausdorff dimension exactly $n-3$, showing that our regularity bound is sharp in every dimension $n\geq3$. The main ingredients in our proof are a generalized Schoen inequality and a corresponding branched sheeting theorem near stationary classical cones and unions of hyperplanes.
CommentsThe proof in Section 3 has been substantially shortened and streamlined. All main results remain unchanged
