Rectifiability of every finite order for stationary integral varifolds
Abstract
We give an alternative proof of a theorem of Brena, De Lellis and Franceschini (arXiv:2503.00649): the support of a stationary integral d dimensional varifold in an open subset of Rn is (ℋd, d) rectifiable of class $(k, α)$ for every positive integer k and every 0 < α< 1, and in particular of class 𝒞∞. Our proof is independent of theirs and proceeds by different means. It compares the varifold, at every scale and at ℋd almost every point, with the graph of a solution of the Euler--Lagrange system of a smoothed area integrand, and thereby derives an excess-decay estimate that is faster than any power of the scale; the conclusion then follows from Santilli's characterisation of higher order rectifiability together with Whitney's extension theorem.
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BibTeX
@misc{kolasinski2026rectifiability,
title = {{Rectifiability of every finite order for stationary integral varifolds}},
author = {Sławomir Kolasiński},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19979},
eprint = {2609.19979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19979},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Rectifiability of every finite order for stationary integral varifolds
AU - Sławomir Kolasiński
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19979
UR - https://arxiv.org/abs/2609.19979
AB - We give an alternative proof of a theorem of Brena, De Lellis and Franceschini (arXiv:2503.00649): the support of a stationary integral d dimensional varifold in an open subset of R^n is (ℋ^d, d) rectifiable of class $(k, α)$ for every positive integer k and every 0 < α< 1, and in particular of class 𝒞^∞. Our proof is independent of theirs and proceeds by different means. It compares the varifold, at every scale and at ℋ^d almost every point, with the graph of a solution of the Euler--Lagrange system of a smoothed area integrand, and thereby derives an excess-decay estimate that is faster than any power of the scale; the conclusion then follows from Santilli's characterisation of higher order rectifiability together with Whitney's extension theorem.
ER -