Regularity of varifolds with bounded anisotropic first variation
Abstract
We prove an ε-regularity theorem for m-varifolds with mean curvature in Lp, p>m, with respect to an anisotropic integrand satisfying a Michael-Simon inequality and the quadratic exposed condition: near sufficiently flat density-one points, such varifolds are representable as C1,α graphs. Combined with the recent proof of the anisotropic Michael-Simon inequality, this establishes an anisotropic Allard regularity theorem in arbitrary codimension for a large class of anisotropic integrands, including those close to the area functional. We also exhibit the first examples of anisotropies satisfying both the uniform scalar atomic condition and the Michael-Simon inequality, that are not close to any ellipsoidal norm. These include the ℓq norms in every dimension and codimension, for explicit ranges of q, and a new class of axisymmetric anisotropies.
Related
- Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below
- Maximum principle for quasilinear SPDE’s on a bounded domain without regularity assumptions
- L2-regularity of minimizers of anisotropic interaction functionals
- Sharp regularity for the anisotropic fractional Laplacian with Morrey data
- Maximal regularity for the Stokes operator in bounded C1 domains
BibTeX
@misc{rosa2026regularity,
title = {{Regularity of varifolds with bounded anisotropic first variation}},
author = {Antonio De Rosa and Benjy Firester and Raphael Tsiamis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20459},
eprint = {2609.20459},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20459},
}
RIS
TY - UNPB
TI - Regularity of varifolds with bounded anisotropic first variation
AU - Antonio De Rosa
AU - Benjy Firester
AU - Raphael Tsiamis
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20459
UR - https://arxiv.org/abs/2609.20459
AB - We prove an ε-regularity theorem for m-varifolds with mean curvature in L^p, p>m, with respect to an anisotropic integrand satisfying a Michael-Simon inequality and the quadratic exposed condition: near sufficiently flat density-one points, such varifolds are representable as C^1,α graphs. Combined with the recent proof of the anisotropic Michael-Simon inequality, this establishes an anisotropic Allard regularity theorem in arbitrary codimension for a large class of anisotropic integrands, including those close to the area functional. We also exhibit the first examples of anisotropies satisfying both the uniform scalar atomic condition and the Michael-Simon inequality, that are not close to any ellipsoidal norm. These include the ℓ^q norms in every dimension and codimension, for explicit ranges of q, and a new class of axisymmetric anisotropies.
ER -