Chapter 22. Tools Used in Proof of Pseudolocality
We know you've got to blame someone for your own confusion.- From "Lunatic Fringe" by Red Rider
This chapter is a companion chapter to the previous chapter on pseu-
dolocality. Here we present the proofs of tools used in the proof of the
pseudolocality theorem. We have formulated some of these tools in more
generality with a view towards their independent interest.
In §1 we give the proofs of the point picking Claims 1 and 2.
In §2 we discuss the convergence of heat kernels for a sequence of solu-
tions to the Ricci flow which converges in the 000 Cheeger-Gromov sense.
In §3 we prove a uniform negative upper bound for the local entropies
centered at the well-chosen bad points at time zero.
In §4 we prove a sharp form of the logarithmic Sobolev inequality, relat-
ing to the isoperimetric inequality.1. A point picking method
We now discuss some space-time point picking methods, which were used
to adjust the sequence of points {(xi, ti)} given by Counterstatement A in
the proof of the pseudolocality Theorem 21.9.
Given a solution to the Ricci flow, the main aim of point picking methods
is to find 'nice' (under the circumstances) sequences of points in space-time
so that one can obtain limits of subsequences of solutions rescaled about
these sequences of points. Roughly speaking, a thematic way in which point
picking is employed is to assume that a curvature estimate (one that we want
to prove) does not hold. This implies there exists a sequence of points with
'large curvature'. From this sequence we wish to obtain a better sequence
of points with large curvature; often 'better' means sufficient to obtain a
contradiction.^1 The purpose of point picking is to accomplish this.
First we describe a general point picking method. This in general helps
us to obtain uniformly bounded curvature at bounded distances away from
the point we have picked. Second we apply point picking to a situation
which will be used in the proof of Theorem 21.9.
(^1) 0ne way of obtaining a contradiction is to show that under the circumstances, small
scales look like large scales, whereas they do not look alike. Note, in this regard, the
dichotomy of no local collapsing and AVR = 0.
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