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Quantitative stratification and global regularity for 1/2-harmonic mappings

Changyu Guo, Guichun Jiang, Changyou Wang, Changlin Xiang, Gaofeng Zheng

Abstract

In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps building on the partial regularity established by Millot-Pegon-Schikorra [Arch. Ration. Mech. Anal. 2021]. Then apply it to show that the set of singular points of such maps can be quantitatively stratified via a new notion of boundary symmetry with the aid of {the celebrated harmonic extension method by Caffarelli-Silverstre}. As in that of Naber-Valtorta, developing the necessary quantitative regularity estimates, and then combining it with the Reifenberg type theorems and a delicate covering argument allow us to get sharp growth estimates on the volume of tubular neighborhood around singular points and establish the rectifiability of each singular stratum.

Quantitative stratification and global regularity for 1/2-harmonic mappings

Abstract

In this paper, we extend the celebrated global regularity theory of Naber-Valtorta [Ann. Math. 2017] to 1/2-harmonic mappings into manifolds. Inspired by their work, we first adapt Lin's defect measure theory [Ann. Math. 1999] to such maps building on the partial regularity established by Millot-Pegon-Schikorra [Arch. Ration. Mech. Anal. 2021]. Then apply it to show that the set of singular points of such maps can be quantitatively stratified via a new notion of boundary symmetry with the aid of {the celebrated harmonic extension method by Caffarelli-Silverstre}. As in that of Naber-Valtorta, developing the necessary quantitative regularity estimates, and then combining it with the Reifenberg type theorems and a delicate covering argument allow us to get sharp growth estimates on the volume of tubular neighborhood around singular points and establish the rectifiability of each singular stratum.
Paper Structure (16 sections, 35 theorems, 209 equations)

This paper contains 16 sections, 35 theorems, 209 equations.

Table of Contents

  1. Introduction and main results
  2. Partial regularity and defect measure
  3. Monotonicity formula and partial regularity
  4. Defect measure theory and stratification of singular sets
  5. Compactness theory of stationary $1/2$-harmonic maps
  6. Quantitative stratification of singular set
  7. Quantitative symmetry and cone splitting principle
  8. Properties of quantitative stratum
  9. Reifenberg theorems and estimates of Jones' number
  10. Covering lemma
  11. An auxiliary covering Lemma
  12. Proof of Lemma \ref{['ti']}
  13. Proof of the main results
  14. Proof of Theorem \ref{['thm80']}
  15. Symmetry implies regularity
  16. ...and 1 more sections

Key Result

Theorem 1.2

Theorems & Definitions (74)

  • Definition 1.1
  • Theorem 1.2: Millot-Sire-15 Millot-Pegon-Schikorra-2021-ARMA
  • Theorem 1.3: Stratification of stationary $1/2$-harmonic maps
  • Definition 1.4: Regularity scale
  • Theorem 1.5: Regularity estimates on minimizing $1/2$-harmonic maps
  • Theorem 1.6: Improved estimates on $1/2$-harmonic maps
  • Lemma 2.1: Millot-Pegon-Schikorra-2021-ARMA
  • Proposition 2.2: Millot-Sire-15
  • Definition 2.3
  • Theorem 2.4: Monotonicity formula, Millot-Pegon-Schikorra-2021-ARMAScheven-2006-MZ
  • ...and 64 more