Index of compact minimal submanifolds of the Berger spheres

TL;DR

This paper investigates the stability and index of compact minimal submanifolds in Berger spheres, revealing conditions for stability and classifying stable cases and minimal surfaces with index one.

Contribution

It provides a detailed analysis of stability conditions and classifies stable minimal submanifolds in Berger spheres, extending understanding beyond the standard sphere case.

Findings

Stable compact minimal submanifolds exist if and only if τ^2 ≤ 1/2.

No stable compact minimal d-dimensional submanifolds when 1/(d+1) < τ^2 ≤ 1.

Classification of stable submanifolds for τ^2 = 1/(d+1).

Abstract

The stability and the index of compact minimal submanifolds of the Berger spheres $\mathbb{S}^{2n+1}_{\tau},\, 0<\tau\leq 1$, are studied. Unlike the case of the standard sphere ($\tau=1$), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if $\tau^2\leq 1/2$. Moreover, there are no stable compact minimal $d$-dimensional submanifolds of $\mathbb{S}^{2n+1}_\tau$ when $1 / (d+1) < \tau^2 \leq 1$ and the stable ones are classified for $\tau^2=1 / (d+1)$ when the submanifold is embedded. Finally, the compact orientable minimal surfaces of $\mathbb{S}^3_{\tau}$ with index one are classified for $1/3\leq\tau^2\leq 1$.