An example of neckpinching for Ricci flow on $S^{n+1}$

We give an example of a class of metrics on $S^{n+1}$ that evolve under the Ricci Flow into a “neckpinch.” We show that the solution has a Type I singularity, and that the length of the neck, i.e.~the region where $|{\mathrm{Rm}}|\sim(T-t)^{-1}$, is bounded from below by $c\sqrt{(T-t)|\log(T-t)|} $ for some $c >0$.

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Published 1 January 2004