- STATEMENT AND INTERPRETATION OF PSEUDOLOCALITY 161
result, one requires a curvature bound on a whole parabolic cylinder in
order to get estimates for the derivatives of curvature. On the other hand,
in the pseudolocality theorem, one has local geometric hypotheses only at
the initial time.
The isoperimetric ratio-condition in Theorem 21.2 is, in a sense, decep-
tively strong. One reason for this is that there is no restriction on the 'scale'
of n (i.e., how small n is) in the ball B (x 0 , r 0 ). For example, curvature
bumps in the sense of Hamilton are ruled out. In particular, it is not pos-
sible for the initial metric g (0) to have the properties that there is a point
Po EB (xo, ro) with B (po, 1/VK) c B (xo, ro) and the smallest eigenvalue
satisfies A.1 (Rm) 2: Kin B (Po, 1/VK).
On the other hand, curvature spikes (i.e., regions of very large curvature
where the size of the region is very small relative to the size of the curva-
ture) are possible, so that the isoperimetric ratio condition does not imply
a curvature bound at the initial time.
Note that in the case of a complete, simply-connected manifold with
constant nonpositive curvature, we have Area(cmr 2: en Vol(nr-^1 (this is
also conjectured for variable nonpositive curvature and has been proven in
dimensions 2, 3, and 4; see Weil [186], Kleiner [109], and Croke [48]). On
the other hand, if the curvature is constant and positive, then
Area(anr i en Vol(nr-^1.
So, roughly speaking, (21.2) rules out too much negative curvature, whereas
(21.3) rules out too much positive curvature.
In §0.2 of [152] Perelman writes:
"In this picture, t corresponds to the scale parameter; the
larger is t, the larger is the distance scale and the smaller
is the energy scale; to compute something on a lower energy
scale one has to average the contributions of the degrees of
freedom, corresponding to the higher energy scale."The statement that the larger t is, the larger is the distance scale is
consistent with the philosophy that Ricci fl.ow makes metrics more homoge-
neous. Given a solution to the Ricci fl.ow, the metric g (t) and its observables
(invariants) at a time t are some sort of average of the metrics and its corre-
sponding observables at earlier times; i.e., the Ricci fl.ow is the 'heat equation
for metrics'.
Given a closed Riemannian manifold (Mn, g ), we may formally define its
distance scale p (g) E [O, oo] to be the square root of the supremum time
of existence of the backward Ricci fl.ow with initial metric g. Intuitively, one
expects that 'most' Riemannian manifolds have distance scale 0. On the
other hand, any Riemannian ma:r;i.ifold which is the time slice of an ancient
solution has distance scale equal to oo. Note that p (cg)= c^112 p (g). It may