 
Summary: AN EXAMPLE OF NECKPINCHING FOR RICCI FLOW ON Sn+1
SIGURD ANGENENT AND DAN KNOPF
Abstract. We give an example of a class of metrics on Sn+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 Rm (T  t)1, is bounded from below by c
p
(T  t) log(T  t) for some c > 0.
1. Preamble
This paper is the first of two in which we study singularity formation in the Ricci flow. As motivation,
consider a solution of the flow
tg = 2 Rc (g)(1a)
g (0) = g0(1b)
starting from an arbitrary Riemannian manifold (Mm
, g0). One should not be surprised if (1) becomes
singular in finite time. Indeed, the simple estimate
tR = R + 2 Rc2
R +
2
m
R2
