On stable constant mean curvature surfaces in $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$

We study the stability of immersed compact constant mean curvature (CMC) surfaces without boundary in some Riemannian 3-manifolds, in particular the Riemannian product spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}.$ We prove that rotational CMC spheres in $\mathbb{H}^2\times\mathbb{R}$ are all stable, whereas in $\mathbb{S}^2\times\mathbb{R}$ there exists some value $H_0\approx 0.18$ such that rotational CMC spheres are stable for $H\geq H_0$ and unstable for $0<H<H_0.$ We show that a compact stable immersed CMC surface in $\mathbb{S}^2\times \mathbb{R}$ is either a finite union of horizontal slices or a rotational sphere. In the more general case of an ambient manifold which is a simply connected conformally flat 3-manifold with nonnegative Ricci curvature we show that a closed stable immersed CMC surface is either a sphere or an embedded torus. Under the weaker assumption that the scalar curvature is nonnegative, we prove that a closed stable immersed CMC surface has genus at most three. In the case of $\mathbb{H}^2\times \mathbb{R}$ we show that a closed stable immersed CMC surface is a rotational sphere if it has mean curvature $H\ge 1/\sqrt {2}$ and that it has genus at most one if $1/\sqrt{3} < H < 1/\sqrt {2}$ and genus at most two if $H=1/\sqrt{3}.$