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

The Ricci flow of the `RP3 geon' and noncompact manifolds with essential minimal spheres