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Optimal higher regularity for biharmonic maps via quantitative stratification

Chang-Yu Guo, Gui-Chun Jiang, Chang-Lin Xiang, Gao-Feng Zheng

TL;DR

This work extends the quantitative stratification framework to biharmonic maps, refining Breiner–Lamm’s regularity results by proving an optimal regularity theory for minimizing biharmonic maps and sharp volume/rectifiability estimates for the quantitative strata of stationary biharmonic maps. Central to the approach are the refined monotonicity formulas, defect-measure analysis, and tangent-measure structure, together with two multiscale covering lemmas and Reifenberg-type arguments that convert energy pinching into symmetry and flatness. The results yield (i) Vol(T_r(S^k_{ε,r}(u))∩B_1)≤C_ε r^{m−k} and rectifiability of S^k_{ε}(u), (ii) optimal higher-regularity for minimizing biharmonic maps with L^{5/ℓ}_{weak} control on derivatives up to order four, and (iii) improved integrability results under the absence of certain biharmonic spheres. Overall, the paper bridges the biharmonic regularity theory with modern quantitative stratification techniques, delivering sharp, geometry-driven regularity conclusions with rigorous multiscale analysis.

Abstract

This little note is devoted to refining the almost optimal regularity results of Breiner and Lamm \cite{Breiner-Lamm-2015} on minimizing and stationary biharmonic maps via the powerful quantitative stratification method introduced by Cheeger and Naber \cite{Cheeger-Naber-2013} and further developed by Naber and Valtorta \cite{Naber-V-2017,Naber-V-2018} for harmonic maps. In particular, we obtain an optimal regularity results for minimizing biharmonic maps.

Optimal higher regularity for biharmonic maps via quantitative stratification

TL;DR

This work extends the quantitative stratification framework to biharmonic maps, refining Breiner–Lamm’s regularity results by proving an optimal regularity theory for minimizing biharmonic maps and sharp volume/rectifiability estimates for the quantitative strata of stationary biharmonic maps. Central to the approach are the refined monotonicity formulas, defect-measure analysis, and tangent-measure structure, together with two multiscale covering lemmas and Reifenberg-type arguments that convert energy pinching into symmetry and flatness. The results yield (i) Vol(T_r(S^k_{ε,r}(u))∩B_1)≤C_ε r^{m−k} and rectifiability of S^k_{ε}(u), (ii) optimal higher-regularity for minimizing biharmonic maps with L^{5/ℓ}_{weak} control on derivatives up to order four, and (iii) improved integrability results under the absence of certain biharmonic spheres. Overall, the paper bridges the biharmonic regularity theory with modern quantitative stratification techniques, delivering sharp, geometry-driven regularity conclusions with rigorous multiscale analysis.

Abstract

This little note is devoted to refining the almost optimal regularity results of Breiner and Lamm \cite{Breiner-Lamm-2015} on minimizing and stationary biharmonic maps via the powerful quantitative stratification method introduced by Cheeger and Naber \cite{Cheeger-Naber-2013} and further developed by Naber and Valtorta \cite{Naber-V-2017,Naber-V-2018} for harmonic maps. In particular, we obtain an optimal regularity results for minimizing biharmonic maps.
Paper Structure (15 sections, 34 theorems, 286 equations)

This paper contains 15 sections, 34 theorems, 286 equations.

Table of Contents

  1. Introduction and main results
  2. Partial regularity and defect measure
  3. Monotonicity formula and partial regularity
  4. Defect measures
  5. Quantitative $\varepsilon$-regularity theorem
  6. Quantitative symmetry and cone splitting principle
  7. Properties of quantitative stratum
  8. Reifenberg theorems and estimates of Jones' number
  9. Covering lemma
  10. The first covering lemma
  11. The second covering lemma
  12. Proofs of main theorems
  13. Proof of Theorem \ref{['thm80']}
  14. Proof of Theorem \ref{['thm84']} and Theorem \ref{['thm85']}
  15. Classical stratification of singular set

Key Result

Theorem 1.2

Let $u\in H_\Lambda^2(B_3(0),N)$ be a stationary biharmonic map, where $H_\Lambda$ is defined in Definition dddd. Then for each $\varepsilon>0$ there exists $C_\varepsilon=C_\varepsilon(m,N,\Lambda,\varepsilon)$ such that for all $r\in(0,1]$, Consequently, for all $r\in(0,1]$, Moreover, for each $k$, $S^k_\varepsilon(u)$ and $S^k(u)$ are $k$-rectifiable and upper Ahlfors $k$-regular, and for ${\

Theorems & Definitions (76)

  • Definition 1.1
  • Theorem 1.2: Stratification of stationary biharmonic maps
  • Definition 1.3: Regularity scale
  • Theorem 1.4: Regularity estimates on minimizing biharmonic maps
  • Theorem 1.5: Improved estimates on biharmonic maps
  • Proposition 2.1: Wang-2004-CPAMAngelsberg-2006
  • Remark 2.2
  • Theorem 2.3: Monotonicity formula, Angelsberg-2006Chang-W-Y-1999
  • Lemma 2.4
  • proof
  • ...and 66 more