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

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

2010-04-11

arxiv.org/abs/1004.1833v1

Mathematics

Differential Geometry

Scientific paper

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.

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