On stability and isoperimetry of constant mean curvature spheres of $${\mathbb {H}}^n\times {\mathbb {R}}$$ and $${\mathbb {S}}^n\times {\mathbb {R}}$$

We approach the one-parameter family of rotational constant mean curvature (CMC) spheres of Hn × R and Sn × R focusing on their stability and isoperimetry properties. We prove that all rotational CMC spheres of Hn × R are stable, and that the ones in Sn × R with sufficiently small (resp. large) mean curvature are unstable (resp. stable). We also show that there exists a one-parameter family of stable CMC rotational spheres in Sn × R which are not isoperimetric (i.e., they do not bound isoperimetric regions). We establish the uniqueness of the regions enclosed by the rotational CMC spheres of Hn × R as solutions to the isoperimetric problem, filling in a gap in the original proof given by Hsiang and Hsiang. We establish, as well, a sharp upper bound for the volume of the spherical regions of Sn × R which are unique solutions to the isoperimetric problem. In essence, all these results come from the fact that the rotational CMC spheres of Hn × R, and those of Sn × R with sufficiently large mean curvature, are nested.