ON STABLE CONSTANT MEAN CURVATURE SURFACES IN S2 x R AND H2 x 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 S-2 x R and H-2 x R. We prove that rotational CMC spheres in H-2 x R are all stable, whereas in S-2 x R there exists some value H-0 approximate to 0.18 such that rotational CMC spheres are stable for H >= H-0 and unstable for 0 < H < H-0. We show that a compact stable immersed CMC surface in S-2 x 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 H-2 x R we show that a closed stable immersed CMC surface is a rotational sphere if it has mean curvature H >= 1/root 2 and that it has genus at most one if 1/root 3 < H < 1/root 2 and genus at most two if H = 1/root 3.