254 9. TYPE I SINGULARITIES
possible pattern, (0, 0, 0, ), may be ruled out by choosing a dilation sequence
so that (8.9) holds, thereby ensuring that the model one obtains is not fiat.
The third observation is the fact that a strong maximum principle holds
for tensors. (See Chapter 4.) Because the curvature operator of any singu-
larity model (M~,9 00 (t) ,x 00 ) is nonnegative at t = 0 and has either the
signature ( + , + , +) or else (0, 0, +) at the single point (x 00 , 0), the strong
maximum principle says that the curvature operator must possess either the
signature ( + , + , + ) or else (0, 0, +) respectively at all points x E M~ and
times t > 0 such that the limit solution 900 (t) exists.
The standard example of a singularity of signature ( +, + , + ) in dimen-
sion 3 is the self-similar solution formed by the shrinking round 3-sphere
(9.1) 4 (T - t) 9can,which we encountered back in Section 5. Singularities of signature (0, 0, + )
are expected to resemble the self-similar 'cylinder solution'
(9.2) ds^2 + 2 ( T - t) 9can
on JR x 52. We saw rigorous examples of such singularities and their asymp-
totic behaviors in Section 5.
When viewed at an appropriate length scale, the geometry of a solution
to the Ricci flow ( M^3 , 9 ( t)) that becomes singular at time T < oo closely
resembles the singularity model (M~, 900 (t), x 00 ) one obtains by blowing-
up the singularity, at least for points near the singularity and times just
before T. The intuitions outlined above, therefore, support the following
heuristic picture. This picture is intended as a guide to our intuition in the
remainder of this chapter.
(1) If a solution (M^3 , 9 (t)) of the unnormalized Ricci flow encounters
a Type I singularity, then after performing dilations correctly and
obtaining an injectivity radius estimate, one expects to see a limit
which is a quotient of either
(a) a compact shrinking round sphere (5^3 ,9(t)), where 9(t) is
given by (9.1), or
(b) a noncom pact shrinking cylinder (JR x 52 , 9 ( t)), where 9 ( t) is
given by (9.2).
(2) If a solution (M^3 , 9 (t)) of the unnormalized Ricci flow encounters
a Type Ila singularity, then after performing dilations correctly
and obtaining an injectivity radius estimate, one expects to see a
quotient of one of the following noncompact limits:
(a) a steady self-similar solution (JR^3 , 9 ( t)) of positive sectional
curvature,
(b) a shrinking cylinder (JR x 52 , 9 ( t)) as in Case 1 b above, or