  It is well-known that the Ricci flow of a closed 3-manifold containing an
essential minimal 2-sphere will fail to exist after a finite time. Conversely,
the Ricci flow of a complete, rotationally symmetric, asymptotically flat
manifold containing no minimal spheres is immortal. We discuss an intermediate
case, that of a complete, noncompact manifold with essential minimal
hypersphere. For 3-manifolds, if the scalar curvature vanishes on asymptotic
ends and is bounded below initially by a negative constant (that depends on the
initial area of the minimal sphere), we show that a singularity develops in
finite time. In particular, this result applies to asymptotically flat
manifolds, which are a boundary case with respect to the neckpinch theorem of M
Simon. We provide numerical evolutions to explore the case where the initial
scalar curvature is less than the bound.
