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arXiv · 2608.13726

Normal curvature of immersed tori of dimension at most $18$

Matteo Raffaelli
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Abstract

For $n\leq 18$, we prove that any smooth immersion of the $n$-torus into the closed unit ball in $\mathbb R^q$ 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 $n\geq19$.

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Matteo Raffaelli. 2026-09-15. Normal curvature of immersed tori of dimension at most $18$. https://arxiv.org/abs/2608.13726

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