2026/07/31 by Gaoming Wang, Xuwen Zhang
Mathematics · #math.AP #math.DG #msc:49Q15 #msc:49Q20 #msc:53C42
60 pages, 1 figure. All comments are welcome
arxiv created 2026/07/31 · arxiv published 2026/07/31 · arxiv updated 2026/08/03
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 Hn-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≥4. We also construct a non-flat stable minimal cone in \mathbb R4 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≥3. 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.