Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold
Abstract
We derive an interior planarity estimate for space-time proper mean curvature flows of n-dimensional submanifolds of ℝn+p, n≥5, that are uniformly quadratically pinched up to the Andrews-Baker threshold 4/3n. The estimate is parabolically scale-invariant and it requires neither curvature nor entropy bounds. The proof combines Naff's gradient estimates with a localization argument in the spirit of Ecker and Huisken. In dimensions five and six, Naff's estimates alone do not give the gradient terms a sign. We bound these terms from below with an argument we developed in a previous work. As a consequence, for n≥5, and by known results for 2≤ n≤4, every space-time proper ancient solution that is uniformly 4/3n-pinched is either a static affine n-plane or lies in a fixed affine (n+1)-dimensional subspace. This replaces the finite entropy and polynomial curvature growth assumptions of Lee, Naff and Zhu by space-time properness, and improves their pinching constant in dimensions two, five and six. For closed flows, we extend Lynch's planarity estimate to a larger range of exponents.
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BibTeX
@misc{ruatta2026planarity,
title = {{Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold}},
author = {Francesco Ruatta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39877},
eprint = {2610.39877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39877},
}
RIS
TY - UNPB
TI - Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold
AU - Francesco Ruatta
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39877
UR - https://arxiv.org/abs/2610.39877
AB - We derive an interior planarity estimate for space-time proper mean curvature flows of n-dimensional submanifolds of ℝ^n+p, n≥5, that are uniformly quadratically pinched up to the Andrews-Baker threshold 4/3n. The estimate is parabolically scale-invariant and it requires neither curvature nor entropy bounds. The proof combines Naff's gradient estimates with a localization argument in the spirit of Ecker and Huisken. In dimensions five and six, Naff's estimates alone do not give the gradient terms a sign. We bound these terms from below with an argument we developed in a previous work. As a consequence, for n≥5, and by known results for 2≤ n≤4, every space-time proper ancient solution that is uniformly 4/3n-pinched is either a static affine n-plane or lies in a fixed affine (n+1)-dimensional subspace. This replaces the finite entropy and polynomial curvature growth assumptions of Lee, Naff and Zhu by space-time properness, and improves their pinching constant in dimensions two, five and six. For closed flows, we extend Lynch's planarity estimate to a larger range of exponents.
ER -