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

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 $n\geq19$.