Precise asymptotics of the Ricci flow neckpinch

The best known finite-time local Ricci flow singularity is the neckpinch, in which a proper subset of the manifold becomes geometrically close to a portion of a shrinking cylinder.In this paper, we prove precise asymptotics for rotationally-symmetric Ricci flow neckpinches.We then compare these rigorous results with formal matched asymptotics for fully general neckpinch singularities. 3.6An upper bound for the spectrum of the linearization 826 3.7 A decomposition into invariant subspaces 828 3.8 The spectrum of the linearization 830 3.9 Our Ansatz 832 3.10 The inner layer |x| = o ( √ τ ) 835 3.11 The intermediate layer |x| = O ( √ τ ) 838 3.12 The outer layer |x| = o 1/ √ Tt