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

Abstract

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 $x:\Sigma^m \rightarrow \mathbb{S}^n(1)$, ($m \leq n$), and by the minimal Riemannian submersions $\pi: \Sigma^m \rightarrow \mathbb{S}^n(1)$, ($m \geq n$).