The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than 1

2023/04/23 by De Lellis, Camillo, Skorobogatova, Anna · 2 citations
#35B65 #35J47 #49N60 #49Q05 #49Q15 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.11555

Abstract

We consider an area-minimizing integral current T of codimension higher than 1 in a smooth Riemannian manifold Σ. In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone according to a real parameter, which we refer to as ``singularity degree''. This parameter determines the infinitesimal order of contact at the point in question between the ``singular part'' of T and its ``best regular approximation''. In this paper we show that the set of points for which the singularity degree is strictly larger than 1, is (m-2)-rectifiable. In a subsequent work we prove that the remaining flat singular points form an (m-2)-null set, thus concluding that the singular set of T is m-2-rectifiable.

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• The Hasse Principle for Geometric Variational Problems: An Illustration via Area-minimizing Submanifolds
• Structure of two-dimensional mod(q) area-minimizing currents near flat singularities: the codimension one case

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