1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

166 21. PERELMAN'S PSEUDOLOCALITY THEOREM

where r can be made as small as we like compared to ro. Now

using the control over the scalar curvature and isoperimetric

constant in B(xoro), we can obtain a contradiction to the

logarithmic Sobolev inequality.'

Below we give a technical summary of the proof. The reader may wish

to return to this after reading the detailed proof which begins in §2.

1.5.3. A summary of the proof of pseudolocality.

The method of proof of the pseudolocality theorem is by contradiction.

Assuming that the theorem is false, there exists a sequence of counterexam-

ples (Mf,gi (t) ,Xoi) corresponding to Oi---+ o+ and CQi---+ o+ (see Counter-

statement A in subsection 2.1). One picks large curvature points (xi, ti) 'not

far from (xoi, O)' and one considers the corresponding adjoint heat kernels

Hi centered at (xi, ti) and their associated nonpositive entropy integrands

Vi = h(2L'.lfi - IV' fil^2 + R 9 i) + fi - n] Hi :S 0,

where fi is related to Hi by Hi= (47rTi)-nl^2 e-li with Ti= ti-t· The points

(xi, ti) are 'well-chosen' in the sense that one has good enough semi-global

curvature control to extract a complete limit of dilations of the solutions

9i (t) based at (xi, ti)·

If the pseudolocality theorem is not true (i.e., if we do not have a suit-

able local curvature bound), then the sequence of Riemannian manifolds

(Mi, 9i (0)), with isoperimetric constants In (gi(O)) = (1 - Oi)cn, has the

following property. Let fli (0) ~ (2ti)-^1 9i (0). Then there exist functions 1/Ji

with supp'l/Ji c B{h(o) ( xoi, (2ti)-^1!^2 ) such that JMi 1/Jf dμ{Ji(O) :::; 1 and

(21.8)

{ ( 21\71/JilEi(O) -1/Jf log ( 1/Jf) - Sn'l/Jf) dμgi(O) :S -/3 :S -/3 { 1/Jf dμ{h(O)

}Mi }Mi

for some f3 independent of i. This contradicts the logarithmic Sobolev in-

equality (22.102) applied to fli (0) on Bg.(o) ( Xoi, (2ti)-^1 l^2 ) since the In (gi (0))

tend to their Euclidean value Cn·
In regards to the hypotheses of the pseudolocality theorem we make
the following remarks. The lower bound on the scalar curvature on the
initial ball is used to compare Perelman's entropy with the almost Euclidean
logarithmic Sobolev inequality; see (22.83). The a/t term in the curvature
estimate is used to obtain curvature control backwards in time from (xi, ti)
on a (scaled) time interval of length bounded from below; see Claim 2 in §1
of Chapter 22 as well as (21.26) and (21.40).

2. Setting up the proof by contradiction and point picking

We shall prove the following theorem, i.e., the r 0 = 1 case of Theorem

21.2. Observe that Theorem 21.2 follows directly from this theorem by
scaling the solution of the Ricci flow (see Remark 21.5).