Index of compact minimal submanifolds of the Berger spheres

The Berger spheres are odd dimensional spheres endowed with a deformation by a parameter $\tau$ of the standard metric in the direction of the Hopf fibers. In this talk we will study the stability and the index of compact minimal submanifolds of the Berger spheres with parameter $\tau\leq 1$. Unlike the case of the standard sphere (that corresponds to the parameter $\tau = 1$), 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 the Berger spheres when $1/(d+1)<\tau^2\leq 1$ and the stable ones are classified for $τ^2=1/(d+1)$ when the submanifold is embedded. If time allows, we will also mentioned the particular case of the 3-dimensional Berger sphere where the compact orientable minimal surfaces with index one are classified for $1/3\leq \tau^2 \leq 1$. This is a joint work with F. Urbano and the paper is available in arXiv:2110.08027 [math.DG]