Structure of two-dimensional mod(q) area-minimizing currents near flat singularities: the codimension one case

2025/06/21 by Skorobogatova, Anna, Spolaor, Luca, Stuvard, Salvatore
#49Q05 #49Q15 #49Q20 #53A10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.17813

Abstract

We obtain a fine structural result for two-dimensional mod(q) area-minimizing currents of codimension one, close to flat singularities. Precisely, we show that, locally around any such singularity, the current is a C^1,α-perturbation of the graph of a radially homogeneous special multiple-valued function that arises from a superposition of homogeneous harmonic polynomials. Additionally, as a preliminary step towards an analogous result in arbitrary codimension, we prove in general that the set of flat singularities of density (q)/(2), where the current is ``genuinely mod(q)", consists of isolated points.

Citations

• Fine Structure of Singularities in Area-Minimizing Currents Mod(q)
• Homologically area-minimizing surfaces mod v have at worst codimension 2 singular sets asymptotically
• Rectifiability of flat singular points for area-minimizing mod(2Q) hypercurrents
• The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than 1
• The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points
• Fine structure of the singular set of area minimizing hypersurfaces modulo p
• Fine properties of branch point singularities: stationary two-valued graphs and stable minimal hypersurfaces near points of density < 3
• An upper Minkowski bound for the interior singular set of area minimizing currents
• Area minimizing hypersurfaces modulo p: a geometric free-boundary problem
• Regularity theory for 2-dimensional almost minimal currents III:\n blowup
• Regularity theory for 2-dimensional almost minimal currents I:\n Lipschitz approximation
• A general regularity theory for stable codimension 1 integral varifolds

Related

• Fine Structure of Singularities in Area-Minimizing Currents Mod(q)
• Flat interior singularities for area almost-minimizing currents
• Fine structure of the singular set of area minimizing hypersurfaces modulo p
• Rectifiability of flat singular points for area-minimizing mod(2Q) hypercurrents
• The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points