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The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points

Camillo De Lellis, Anna Skorobogatova

TL;DR

The paper introduces a robust notion of singularity degree for flat singular points of area-minimizing integral currents in codimension at least two, defined via the limit homogeneity (singular frequency) of fine blow-ups around a flat tangent cone. It proves that this singularity degree is well-defined (independent of blow-up sequence) and that all fine blow-ups are homogeneous of this degree, with a lower bound of 1 provided by a Hardt–Simon-type inequality. Moreover, when the degree exceeds 1, the tangent cone is unique and the current decays to it at a polynomial rate, while a frequency- BV framework yields precise control on the variation of the frequency across scales. This groundwork supports two companion works showing (m-2)-rectifiability of the flat singular set and almost everywhere uniqueness of tangent cones, and it contrasts with KW’s planar-frequency approach by leveraging center manifolds and Dir-minimizing multi-valued maps. The results collectively refine our understanding of the singular structure in higher codimension and lay a path toward full rectifiability and uniqueness conclusions for the singular set.

Abstract

We consider an area-minimizing integral current of dimension $m$ and codimension at least $2$ and fix an arbitrary interior singular point $q$ where at least one tangent cone is flat. For any vanishing sequence of scales around $q$ along which the rescaled currents converge to a flat cone, we define a suitable singularity degree of the rescalings, which is a real number bigger than or equal to $1$. We show that this number is independent of the chosen sequence and we prove several interesting properties linked to its value. Our study prepares the ground for two companion works, where we show that the singular set is $(m-2)$-rectifiable and the tangent cone is unique at $\mathcal{H}^{m-2}$-a.e. point.

The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points

TL;DR

The paper introduces a robust notion of singularity degree for flat singular points of area-minimizing integral currents in codimension at least two, defined via the limit homogeneity (singular frequency) of fine blow-ups around a flat tangent cone. It proves that this singularity degree is well-defined (independent of blow-up sequence) and that all fine blow-ups are homogeneous of this degree, with a lower bound of 1 provided by a Hardt–Simon-type inequality. Moreover, when the degree exceeds 1, the tangent cone is unique and the current decays to it at a polynomial rate, while a frequency- BV framework yields precise control on the variation of the frequency across scales. This groundwork supports two companion works showing (m-2)-rectifiability of the flat singular set and almost everywhere uniqueness of tangent cones, and it contrasts with KW’s planar-frequency approach by leveraging center manifolds and Dir-minimizing multi-valued maps. The results collectively refine our understanding of the singular structure in higher codimension and lay a path toward full rectifiability and uniqueness conclusions for the singular set.

Abstract

We consider an area-minimizing integral current of dimension and codimension at least and fix an arbitrary interior singular point where at least one tangent cone is flat. For any vanishing sequence of scales around along which the rescaled currents converge to a flat cone, we define a suitable singularity degree of the rescalings, which is a real number bigger than or equal to . We show that this number is independent of the chosen sequence and we prove several interesting properties linked to its value. Our study prepares the ground for two companion works, where we show that the singular set is -rectifiable and the tangent cone is unique at -a.e. point.
Paper Structure (33 sections, 33 theorems, 265 equations, 1 figure)

This paper contains 33 sections, 33 theorems, 265 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Flat singularities
  3. Singularity degree
  4. Comparison with the work of Krummel & Wickramasekera
  5. Main statements
  6. Compactness procedure
  7. Main results
  8. Rectifiability
  9. Frequency function
  10. Proof of Lemma \ref{['l:finite-vs-infinite']}
  11. Comparison of this article with KW
  12. The Hardt-Simon inequality and coarse blow-ups
  13. Coarse blow-ups
  14. Closure under rescalings
  15. Proof of Theorem \ref{['thm:HS']}
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $T$ be an $m$-dimensional area-minimizing current in a $C^{3,\kappa_0}$ complete Riemannian manifold of dimension $m+\bar{n}\geqslant m+2$, with $\kappa_0>0$. Then $\mathrm{Sing} (T)$ is $(m-2)$-rectifiable and there is a unique tangent cone at $\mathcal{H}^{m-2}$-a.e. $q\in \mathrm{Sing} (T)$.

Figures (1)

  • Figure :

Theorems & Definitions (55)

  • Theorem 1.1
  • Example 1.2
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9
  • Theorem 2.10
  • Theorem 2.11
  • ...and 45 more