PreprintCC BY 4.0arXiv
The optimal isoperimetric inequality for varifolds
Abstract
We prove the Brendle isoperimetric inequality for varifolds of finite mass and finite total first variation, allowing an arbitrary singular part. The distributional mean curvature is assumed only integrable. The result applies to integral varifolds and, more generally, to rectifiable varifolds with a positive lower density bound and weight measure concentrated on countably many C2 submanifolds. The proof uses optimal transport and a locality theorem for the tangential Laplacian of an ambient convex function. The isoperimetric constant is sharp in codimensions one and two.
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BibTeX
@misc{santilli2026optimal,
title = {{The optimal isoperimetric inequality for varifolds}},
author = {Mario Santilli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24716},
eprint = {2609.24716},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24716},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The optimal isoperimetric inequality for varifolds
AU - Mario Santilli
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24716
UR - https://arxiv.org/abs/2609.24716
AB - We prove the Brendle isoperimetric inequality for varifolds of finite mass and finite total first variation, allowing an arbitrary singular part. The distributional mean curvature is assumed only integrable. The result applies to integral varifolds and, more generally, to rectifiable varifolds with a positive lower density bound and weight measure concentrated on countably many C^2 submanifolds. The proof uses optimal transport and a locality theorem for the tangential Laplacian of an ambient convex function. The isoperimetric constant is sharp in codimensions one and two.
ER -