Fat equator effect and Minimality in immersions and submersions of the Sphere

Inspired by the equatorial concentration of measure phenomenon in the sphere, a result which is deduced from the general, (and intrinsic), concentration of measure in ${\mathbb{S}}^n(1)$, we describe in this paper an equatorial concentration of measure satisfied by the closed, (compact without boundary), isometric and minimal immersions $x:{\mathrm{\Sigma }}^m\to {\mathbb{S}}^n(1)$, ($m\le n$), and by the minimal Riemannian submersions $\pi :{\mathrm{\Sigma }}^m\to {\mathbb{S}}^n(1)$, ($m\ge n$).