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Quantitative stratification and optimal regularity for harmonic almost complex structures

Chang-Yu Guo, Ming-Lun Liu, Chang-Lin Xiang

TL;DR

This work delivers a near-complete optimal regularity theory for energy-minimizing harmonic almost complex structures by unveiling a divergence-free structure that enables a streamlined partial regularity proof. It adapts the quantitative stratification framework of Naber–Valtorta to the MHACS setting, obtaining sharp volume bounds for singular strata and proving their rectifiability via Reifenberg-type arguments. The results include optimal regularity estimates and a bound on the singular set dimension, matching the sharpness seen in the harmonic-map setting and suggesting avenues for extending to higher-order (biharmonic/polyharmonic) structures. Overall, the paper advances the understanding of regularity for geometric variational problems linked to almost complex structures and provides tools potentially applicable to broader nonlinear elliptic systems.

Abstract

In a recent interesting work [15], W.Y. He established the important partial regularity theory and the almost optimal higher regularity theory for energy minimizing harmonic almost complex structures. Based on a new observation on the structure of equations, we give an easier new proof of the partial regularity theorem, and adapting the powerful quantitative stratification method of Naber-Valtorta [22], we further prove the rectifiability of singular stratum of energy minimizing harmonic almost complex structures. Based on this, we establish an optimal regularity theory, which improves the corresponding result of He.

Quantitative stratification and optimal regularity for harmonic almost complex structures

TL;DR

This work delivers a near-complete optimal regularity theory for energy-minimizing harmonic almost complex structures by unveiling a divergence-free structure that enables a streamlined partial regularity proof. It adapts the quantitative stratification framework of Naber–Valtorta to the MHACS setting, obtaining sharp volume bounds for singular strata and proving their rectifiability via Reifenberg-type arguments. The results include optimal regularity estimates and a bound on the singular set dimension, matching the sharpness seen in the harmonic-map setting and suggesting avenues for extending to higher-order (biharmonic/polyharmonic) structures. Overall, the paper advances the understanding of regularity for geometric variational problems linked to almost complex structures and provides tools potentially applicable to broader nonlinear elliptic systems.

Abstract

In a recent interesting work [15], W.Y. He established the important partial regularity theory and the almost optimal higher regularity theory for energy minimizing harmonic almost complex structures. Based on a new observation on the structure of equations, we give an easier new proof of the partial regularity theorem, and adapting the powerful quantitative stratification method of Naber-Valtorta [22], we further prove the rectifiability of singular stratum of energy minimizing harmonic almost complex structures. Based on this, we establish an optimal regularity theory, which improves the corresponding result of He.

Paper Structure

This paper contains 20 sections, 32 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Partial regularity and compactness theory
  5. Monotonicity formula and partial regularity
  6. Compactness property
  7. Unique continuation property
  8. Tangent almost complex structure
  9. Quantitative $\varepsilon$-regularity theorem
  10. Quantitative symmetry and cone splitting principle
  11. Basic properties of quantitative stratum
  12. Reifenberg theorems and estimates of Jones' number
  13. Covering lemma
  14. Proof of Lemma \ref{['lemma: main covering lemma']}
  15. Proof of Lemma \ref{['lemma: covering lemma I']}
  16. ...and 5 more sections

Key Result

Theorem A

Let $J\in W^{1,2}(\mathcal{J}_g(M))$ be a minimizing harmonic almost complex structure. There exists $\varepsilon=\varepsilon(m, M, g)>0$ such that for any $p\in M$ if, for some $r\in(0,1)$ we have then $J\in C^{\infty}(B_{\frac{r}{2}}(p))$.

Theorems & Definitions (68)

  • Definition 1.1: Harmonic almost complex structures
  • Theorem A: He-2019
  • Theorem B: He-2019
  • Definition 1.2: Regularity scale
  • Theorem 1.3: Optimal volume estimates and rectifiability for singular strata
  • Theorem 1.4: Regularity estimates on minimizing harmonic almost complex structure
  • Proposition 2.1: He-2019
  • Theorem 2.2: Monotonicity formula
  • proof
  • Remark 2.3
  • ...and 58 more