1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

212 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

COROLLARY 22.21 (Isometry group is preserved under Ricci fl.ow). If

(Mn,g(t)), t E [O,T], is a complete solution to the Ricci flow with bounded

curvature, then Isom (g ( t)) = Isom (g ( 0)) for all t > 0.

Indeed, Theorem 22.20 implies that Isom (g (t)) c Isom (g (0)) fort> 0,

whereas the uniqueness theorems of Hamilton (for closed manifolds) and

Chen-Zhu [37] (for complete solutions with bounded curvature on noncom-

pact manifolds) imply Isom (g (0)) C Isom (g (t)) fort> 0.

Recall that, by result of Banda [10], a complete solution (Mn, g (t)),

t E [O, T), to the Ricci fl.ow with bounded curvature is real analytic in the

space variables for any t E (0, T) (see §2 of Chapter 13 in Part II). This

implies that if (Mf,gi (t)), i = 1,2, are complete solutions with bounded

curvature such that gl ( t') and g2 ( t') restricted to some pair of open sets

are isometric for some t' E (0, T), then the universal covers with the lifted

metrics (M?,g1 (t')) and (M~,g2 (t')) are isometric (see the original §4 of

Myers [136] or Corollary 6.4 of Kobayashi and Nomizu [111]; the case of

Einstein metrics is Corollary 5.28 in Besse [15]). By forward and backward

uniqueness, this implies 91 (t) and 92 (t) are isometric for all t E [O, T).

Regarding the property of unique continuation, one may also ask the

following question. Can one define a 'canonical' notion of solution of the

Ricci fl.ow with surgery on closed 3-manifolds whose properties include the

following?

CRITERION 22.22 (For canonical Ricci fl.ow with surgery on closed 3-man-

ifolds). :If (Mr (t) , gl (t)) and (M~ (t), g2 (t)) are solutions of the Ricci flow

with surgery on the time interval [O, T], where the manifolds Mr (t) and

M~ (t) are closed at all nonsurgery times, where Mr (0) and M~ (0) are

connected, and such that for some t' E (0, T] we have that gl (t') and g2 (t')

restricted to some pair of open sets are isometric, then (Mr (t), g 1 (t)) and

(M~ (t) ,g2 (t)) must be isometric for all t E [O,T]. (Note that gl and g2

are singular metrics at the surgery times.) '

This would have the following consequence. If Ricci fl.ow with surgery

evolves a dumbbell shaped 3-sphere into two (disjoint) 3-spheres with a

single surgery occurring at some time t* E (0, T), then the metric at any time

t > t* on an open subset of one of the two 3-spheres 'uniquely determines'

the metric at all times after the surgery time on the other 3-sphere.
Finally, we remark that on p. 3 of [152], when speculating on the 'Wilso-
nian picture' of renormalization group (RG) fl.ow, Perelman wrote:

"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. In other

words, decreasing of t should correspond to looking at our