- STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY 163
applies to a time interval much smaller than the time elapsed before the
singularity forms.
We also compare this with the situation of the heat equation on Eu-
clidean space. There, the equation is linear and singularities do not form.
The fundamental solution H (x, y, t) = ( 4nt)-n/^2 e-fx-yf2
/^4 t tends to 6y (x)as t --+ 0 and satisfies the estimate 0 < H ( x, y, t) :S ( 4nt )-n/^2 for all t > 0.
Moreover, for Ix - YI ~ c > 0, we have H (x, y, t) :S (4nt)-n/^2 e-c^2 /4t.
Theorem 21.2 is also similar in some respects to c:-regularity results such
as that of Ecker and Huisken [56] for the mean curvature flow (see §10.6*
of [152]).1.4. Topping's example.
We now give an example (due to Peter Topping [180]) which showsthat Theorem 21.2 is false if we do not assume that (Mn, g (t)) is complete.
Consider the cylinder
51 (r) x [-1, 1]with the flat product metric, where 51 (r) denotes the circle of radius r (i.e.,
circumference 2nr) and r > 0 is small. Topologically we cap each of the
two ends of the cylinder with a disc D^2. Metrically we use a cutoff function
to smoothly blend the cylinder metric with the round hemisphere si ( r) in
thin collars about their boundaries to construct a rotationally symmetric
Riemannian metric g 0 with nonnegative curvature on a surface E^2 which is
diffeomorphic to the 2-sphere. In our construction we may assume
224nr :S Area 90 (E) :S 5nr
provided r is sufficiently small.^4
Let
(E^2 ,gr(t)), tE[O,Tr),
be the maximal solution of the Ricci flow with gr (0) = g 0. We may estimate
the maximal time Tr as follows. By the Gauss-Bonnet theorem we have
.!!:._ Area 9 ,.(t) (E) = - { R 9 ,.(t) dμ 9 ,.(t) = -8n,
dt J~so that
In particular,
(21.5) Tr < 2_ Area 9 r (E) <
11- 81f^0 - 20 r.
In fact, by Hamilton's result that metrics on 52 with nonnegative curvature
shrink to round points under the Ricci flow, we have Tr = i1r Vol 90 (E).
Note that limr-+O+ Tr= 0.
(^4) Note that 47rr =Area (S (^1) (r) x [-1, ll).