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The anisotropic Michael-Simon inequality

  • Benjy Firester
  • Raphael Tsiamis
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Abstract

We prove a Michael-Simon inequality for varifolds of every dimension and codimension with respect to anisotropic energies. In particular, this resolves the problem for every convex even hypersurface anisotropy and, using work of Allard, yields the regularity of hypersurfaces with bounded anisotropic mean curvature in every dimension. By the results of De Philippis and Pigati, it also implies density bounds and compactness properties for rectifiable varifolds with uniformly bounded anisotropic first variation. The proof relies on a projection method for anisotropic stress measures as well as the recent resolution of the vanishing mass conjecture by Gennaioli and Rindler.

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Cite

BibTeXRIS
BibTeX
@misc{firester2026anisotropic,
  title = {{The anisotropic Michael-Simon inequality}},
  author = {Benjy Firester and Raphael Tsiamis},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.31164},
  eprint = {2608.31164},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.31164},
}
RIS
TY  - UNPB
TI  - The anisotropic Michael-Simon inequality
AU  - Benjy Firester
AU  - Raphael Tsiamis
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2608.31164
UR  - https://arxiv.org/abs/2608.31164
AB  - We prove a Michael-Simon inequality for varifolds of every dimension and codimension with respect to anisotropic energies. In particular, this resolves the problem for every convex even hypersurface anisotropy and, using work of Allard, yields the regularity of hypersurfaces with bounded anisotropic mean curvature in every dimension. By the results of De Philippis and Pigati, it also implies density bounds and compactness properties for rectifiable varifolds with uniformly bounded anisotropic first variation. The proof relies on a projection method for anisotropic stress measures as well as the recent resolution of the vanishing mass conjecture by Gennaioli and Rindler.
ER  -