  Let $\Gamma$ be a nondegenerate geodesic in a compact Riemannian manifold
$M$. We prove the existence of a partial foliation of a neighbourhood of
$\Gamma$ by CMC surfaces which are small perturbations of the geodesic tubes
about $\Gamma$. There are gaps in this foliation, which correspond to a
bifurcation phenomenon. Conversely, we also prove, under certain restrictions,
that the existence of a partial CMC foliation of this type about a submanifold
$\Gamma$ of any dimension implies that $\Gamma$ is minimal.
