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

Differential Geometry 2022-12-07 v1 Analysis of PDEs

Abstract

For an intergral 22-varifold V=v(Σ,θ1)V=\underline{v}(\Sigma,\theta_{\ge 1}) in the unit ball B1B_1 passing through the original point, assuming the critical Allard condition holds, that is, the area μV(B1)\mu_V(B_1) is close to the area of a unit disk and the generalized mean curvature has sufficient small L2L^2 norm, we prove Σ\Sigma is bi-Lipschitz homeomorphic to a flat disk in R2\mathbb{R}^2 locally.

Keywords

Cite

@article{arxiv.2212.03043,
  title  = {Bi-Lipschitz Regularity of 2-Varifolds with the Critical Allard Condition},
  author = {Yuchen Bi and Jie Zhou},
  journal= {arXiv preprint arXiv:2212.03043},
  year   = {2022}
}

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72pages