Cauchy-Riemann inequalities on 2-spheres of $\mathbb{R}^7$

TL;DR

This paper establishes a Cauchy-Riemann inequality for functions on 2-spheres in ^7, characterizes equality cases, extends the inequality to 4-tuples, and applies it to geometric stability in ^7.

Contribution

It proves a new integral inequality for functions on 2-spheres in ^7 and explores its implications for geometric stability and eigenfunctions.

Findings

The inequality holds for any smooth functions on ^2.

Equality occurs iff functions are related -eigenfunctions.

2-spheres are not -stable with parallel mean curvature in ^7.

Abstract

We prove that an integral Cauchy-Riemann inequality holds for any pair of smooth functions $(f,h)$ on the 2-sphere $\mathbb{S}^2$, and equality holds iff $f$ and $h$ are related $\lambda_1$-eigenfunctions. We extend such inequality to 4-tuples of functions, only valid on the $L^2$-orthogonal complement of a suitable nonzero finite dimensional space of functions. As a consequence we prove that 2-spheres are not $\Omega$-stable surfaces with parallel mean curvature in $\mathbb{R}^7$ for the associative calibration $\Omega$.