PreprintCC BY 4.0arXiv
The planarity estimate in dimensions five and six
Abstract
Naff's planarity estimate is a crucial tool for analysing singularities of quadratically pinched mean curvature flows in high codimension, showing that high-curvature regions become asymptotically codimension one. Naff established the estimate throughout the full Andrews--Baker pinching range in dimensions at least $7$, while in dimensions $5$ and $6$ a stronger pinching assumption was required. We close this remaining gap, proving the planarity estimate in dimensions $5$ and $6$ throughout the full Andrews--Baker range. The key new ingredient is a sharp Kato-type inequality relating the twisting of the principal normal to the gradient of the second fundamental form.
Related
- The square function estimate for the cone in four dimensions
- Scaling Limit of Critical Loop Soup Clusters in Three, Four, and Five Dimensions
- Blow-up for energy supercritical defocusing nonlinear Schrödinger equations in dimensions three, four and five
- Rigidity of Stationary Navier-Stokes Solutions with Transverse bound in Dimensions Five and Higher
- Fluctuations of the quadratic matching cost on the flat torus: dimensions five and higher
BibTeX
@misc{lynch2026planarity,
title = {{The planarity estimate in dimensions five and six}},
author = {Stephen Lynch},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39330},
eprint = {2609.39330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39330},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The planarity estimate in dimensions five and six
AU - Stephen Lynch
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39330
UR - https://arxiv.org/abs/2609.39330
AB - Naff's planarity estimate is a crucial tool for analysing singularities of quadratically pinched mean curvature flows in high codimension, showing that high-curvature regions become asymptotically codimension one. Naff established the estimate throughout the full Andrews--Baker pinching range in dimensions at least $7$, while in dimensions $5$ and $6$ a stronger pinching assumption was required. We close this remaining gap, proving the planarity estimate in dimensions $5$ and $6$ throughout the full Andrews--Baker range. The key new ingredient is a sharp Kato-type inequality relating the twisting of the principal normal to the gradient of the second fundamental form.
ER -