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

TL;DR

This paper explores how measure concentrates around the equator in spheres for minimal immersions and submersions, extending the classical measure concentration phenomenon to geometric mappings.

Contribution

It introduces an equatorial concentration of measure phenomenon for minimal immersions and submersions in spheres, generalizing intrinsic measure concentration results.

Findings

Describes equatorial measure concentration in minimal immersions

Establishes measure concentration in minimal Riemannian submersions

Extends classical measure concentration to geometric mappings

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 $\pi: \Sigma^m \rightarrow \mathbb{S}^n(1)$, ($m \geq n$).