2024/01/31 by Zhenhua Liu, Liu, Zhenhua
Computer Science · #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2401.18074
openalex publication_date 2024/01/31 · openalex created_date 2024/02/02 · openalex updated_date 2026/07/28
De Lellis and coauthors have proved a sharp regularity theorem for area-minimizing currents in finite coefficient homology. They prove that area-minimizing mod v currents are smooth outside of a singular set of codimension at least 1. Classical examples like triple junctions demonstrate that their result is sharp. Surprisingly, even though their regularity theorem cannot be improved for any fixed v, if one instead fixes the homology class, then v asymptotically one can always achieve more regularity. For any integral homology class [Σ] on any Riemannian manifold, we show that for v large, any area-minimizing mod v current in [Σ\mod v] must be an integral current, thus having a singular set of codimension at least 2 in general and of codimension at least 7 in the hypersurface case. Similar results are obtained for Plateau problems in Euclidean space. Our work is inspired by Morgan's work and based on De Lellis' and coauthors' work.