Chapter 21. Perelman's Pseudolocality Theorem
You speak to me in riddles and you speak to me in rhymes.- From "Possession" by Sarah MacLachlin
One mantra is that the Ricci flow smooths out metrics. Among the
results which support this contention are the following:
(1) The fact that many geometric quantities derived from the metric,
such as the various curvature tensors, satisfy heat-type equations.
See Chapter 6 of Volume One.
(2) Bando's Bernstein-type global derivative estimates, assuming a
bound on the curvature, which provide bounds for tm/^2 l'Vm Rml
for m E N. Here the bounds for the derivatives of curvature im-
prove as t increases at least for short time. See Chapter 7 of Volume
One and Chapter 14 of Part II, respectively.
(3) The long time existence and convergence results in all dimensions,
usually under a positive curvature hypothesis, which exhibit smooth-
ing to a best possible type of metric, namely a constant positive
sectional curvature metric. See Chapters 5 and 6 of Volume One
for the case of dimensions 2 and 3 and see Chapter 11 in Part II
for higher dimensions.
(4) The entropy and reduced volume monotonicity formulas of Perel-
man (we think of monotonicity as evidence of smoothing), which
imply the no local collapsing theorem and affect the classification
of singularity models. See Chapters 6 and 8 in Part I, respectively.
(5) The differential inequalities satisfied by the reduced distance func-
tion and the structure theorems for /'\;-solutions. (Heuristically such
results would not be possible without smoothing.) See Chapters 7
and 8 of Part I.
Recall from Problem 17.33 that one goal in the study of Ricci flow and
geometric analysis is to localize various formulas. The localization of the
aforementioned Bando global derivative estimates is Shi's local derivative
estimates. For the heat equation on a Riemannian manifold with a lower
bound on the Ricci curvature, the Li-Yau differential Harnack estimate was
originally proved in localized form, which yielded a sharp global estimate in
the case of Re 2:: 0 (see Theorem 25.8 below). Another example of a local
estimate for the heat equation is the parabolic mean value inequality (see
Theorem 25.2).
157