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We approach the one-parameter family of rotational constant mean curvature(CMC) spheres of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R$focusing on their stability and isoperimetry properties. We prove that allrotational CMC spheres of $\mathbb H^n\times\mathbb R$ are stable, and that theones in $\mathbb S^n\times\mathbb R$ with sufficiently small (resp.~large) meancurvature are unstable (resp.~stable). We also show that there exists aone-parameter family of stable CMC rotational spheres in $\mathbbS^n\times\mathbb R$ which are not isoperimetric (i.e., they do not boundisoperimetric regions). We establish the uniqueness of the regions enclosed bythe rotational CMC spheres of $\mathbb H^n\times\mathbb R$ as solutions to theisoperimetric problem, filling in a gap in the original proof given by Hsiangand Hsiang. We establish, as well, a sharp upper bound for the volume of thespherical regions of $\mathbb S^n\times\mathbb R$ which are unique solutions tothe isoperimetric problem. In essence, all these results come from the factthat the rotational CMC spheres of $\mathbb H^n\times\mathbb R$, and those of$\mathbb S^n\times\mathbb R$ with sufficiently large mean curvature, arenested.

On Stability and Isoperimetry of Constant Mean Curvature Spheres of $\mathbb H^n\times\mathbb R$ and $\mathbb S^n\times\mathbb R.$