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Homologically area-minimizing surfaces mod $v$ have at worst codimension 2 singular sets asymptotically

Zhenhua Liu

Abstract

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.

Homologically area-minimizing surfaces mod $v$ have at worst codimension 2 singular sets asymptotically

Abstract

De Lellis and coauthors have proved a sharp regularity theorem for area-minimizing currents in finite coefficient homology. They prove that area-minimizing mod currents are smooth outside of a singular set of codimension at least Classical examples like triple junctions demonstrate that their result is sharp. Surprisingly, even though their regularity theorem cannot be improved for any fixed , if one instead fixes the homology class, then asymptotically one can always achieve more regularity. For any integral homology class on any Riemannian manifold, we show that for large, any area-minimizing mod current in must be an integral current, thus having a singular set of codimension at least in general and of codimension at least 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.
Paper Structure (18 sections, 18 theorems, 59 equations)

This paper contains 18 sections, 18 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Sketch of proof
  3. Preliminaries and Lemmas
  4. Manifolds and neighborhoods
  5. Algebraic topology
  6. Integral currents and flat chains
  7. Mod $v$ currents
  8. Mass and comass
  9. Representing homology classes by submanifolds
  10. A priori density bounds
  11. Norms on homology
  12. Regularity theorems
  13. Proof of Theorem \ref{['modvc']},\ref{['modvb']},\ref{['modva']} and Corollary \ref{['setk']}
  14. Proof of the theorems
  15. Proof of Corollary \ref{['setk']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $[\Sigma]$ be a $d$-dimensional integral homology class on a $d+c$-dimensional smooth compact closed Riemannian manifold $M^{d+c}$, with $d,c\ge1.$ Then for $v$ large enough, any area-minimizing mod $v$ current in $[\Sigma\mod v]$ is smooth outside of a singular set of codimension at least $2$ f

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1.3
  • Remark 4
  • Theorem 1.4
  • Remark 5
  • Definition 1.1
  • ...and 36 more