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Normal curvature of immersed tori of dimension at most $18$
Abstract
For n≤ 18, we prove that any smooth immersion of the n-torus into the closed unit ball in ℝq has a point at which the spherical average of lvert II(v,v)rvert2 is at least 3n/(n+2). This answers a question of Petrunin in these dimensions. The proof combines the scalar curvature obstruction for the torus with a conformal Laplacian argument, reducing the problem to a one-dimensional differential inequality. We also show that this reduction cannot yield the result in dimensions n≥19.
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BibTeX
@misc{raffaelli2026normal,
title = {{Normal curvature of immersed tori of dimension at most $18$}},
author = {Matteo Raffaelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13726},
eprint = {2608.13726},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13726},
}
RIS
TY - UNPB
TI - Normal curvature of immersed tori of dimension at most $18$
AU - Matteo Raffaelli
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2608.13726
UR - https://arxiv.org/abs/2608.13726
AB - For n≤ 18, we prove that any smooth immersion of the n-torus into the closed unit ball in ℝ^q has a point at which the spherical average of lvert II(v,v)rvert^2 is at least 3n/(n+2). This answers a question of Petrunin in these dimensions. The proof combines the scalar curvature obstruction for the torus with a conformal Laplacian argument, reducing the problem to a one-dimensional differential inequality. We also show that this reduction cannot yield the result in dimensions n≥19.
ER -