Index of compact minimal submanifolds of the Berger spheres

Abstract

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

arXiv comment37 pages, 1 figure. Typos corrected. Some arguments have been clarified. Accepted for its publication in Calculus of Variations and Partial Differential Equations