Bi-Lipschitz Regularity of 2-Varifolds with the Critical Allard Condition

2022/12/06 by Yuchen Bi, Jie Zhou, Bi, Yuchen +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2212.03043

openalex publication_date 2022/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an intergral 2-varifold V=\underlinev(Σ,θ_{≥ 1}) in the unit ball B₁ passing through the original point, assuming the critical Allard condition holds, that is, the area μ_V(B₁) is close to the area of a unit disk and the generalized mean curvature has sufficient small L² norm, we prove Σ is bi-Lipschitz homeomorphic to a flat disk in ℝ² locally.

Cited by

• Quantitative Stability of the Clifford Torus as a Willmore Minimizer

Related

• Critical Allard regularity: pointwise tilt-excess estimates
• An Allard type regularity theorem for varifolds with Hölder continuous generalized normal
• Allard-Type Regularity for Varifolds with Prescribed Contact Angle
• Allard's interior ε-Regularity Theorem in Alexandrov spaces
• Global regularity of integral 2-varifolds with square integrable mean curvature