2022/09/24 by Nicolau S. Aiex, Aiex, Nicolau S., Sean McCurdy +3
Mathematics · #49Q20 #53A10 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #math.DG #msc:49Q20 #msc:53A10
paper · pdf · doi:10.48550/arxiv.2209.11992
22 pages
arxiv created 2022/09/24 · openalex publication_date 2022/09/24 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28 · arxiv updated 2026/08/06
We prove local measure bounds on the tubular neighbourhood of the singular set of codimension one stationary integral n-varifolds V in Riemannian manifolds which have both: (i) finite index on their smoothly embedded part; and (ii) Hn-1-null singular set. A direct consequence of such a bound is a bound on the upper Minkowski content of the singular set of such a varifold in terms of its total mass and its index. Such a result improves on known bounds, namely the corresponding bound on the Hn-7-measure of the singular set established by A. Song ([10]), as well as the same bounds established by A. Naber and D. Valtorta ([6]) for codimension one area minimising currents. Our results also provide more structural information on the singular set for codimension one integral varifolds with finite index (on the regular part) and no classical singularities established by N. Wickramasekera ([11]).