36 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES
5. Notes and commentary
The monotonicity formulas which are applied in this chapter were dis-
covered partly based on the notion of self-similarity.
On a more philosophical note, we venture to ask the following general
question.
PROBLEM 17.33. In Ricci fl.ow some of the guiding ideas and principles,
among others, used to study the geometry of solutions are
(1) analogies with the heat equation and its smoothing aspects,
(2) applications of the maximum principle and monotonicity,
(3) self-similar solutions, i.e., Ricci solitons, used to find quantities to
.~ estimate,
( 4) natural space-time quantities such as the reduced distance,
(5) point picking and determining location and scale.
Can one discover new guiding principles and ideas?
(A) One goal (espoused by Hamilton) is to localize various formulas.
(Perelman's pseudolocality in effect does this for the curvature evo-
lution under certain hypotheses.) Can one find new quantities on
space-time (or some larger system) to help accomplish this?
(B) An important problem (espoused by Perelman) is to formulate a
notion of weak solution which enables the fl.ow to continue past
singularities.
(C) Largely uncharted territory is the role of Riemannian Ricci fl.ow
in higher dimensions. Note that Lie algebra aspects of the evolu-. tion of the curvature operator, which began with Hamilton's work
on 4-manifolds, have been studied by Bohm and Wilking with ap-
plications to the classification of closed manifolds with 2-positive
curvature operatbr. In addition, Hamilton and Perelman proved a
number of results in arbitrary dimensions in their works.
(D) Is it fruitful to study Ricci fl.ow on spin manifolds or some other
large class of manifolds?
§1. In contrast to the lower bound given by Lemma 17.8 for the volume
of a solution g (t) to Ricci fl.ow with .A (g (t)) ::; 0, we have the following.
Upper bound for the volume of a solution with .A > 0. Suppose that a
solution (Mn, g ( t)) of the Ricci flow on a closed manifold and maximal time
interval [O, T) has .A (g (0)) > 0. Since ft.A (g (t)) 2: ~). (g (t))^2 (see Lemma
5.25 in Part I), we have
1
.A(g(t));::::: .A(g(0))-1-~t'so that T::; ~.A (g (0))-^1 < oo. Since
d.
dt log Vol (g (t)) = -Ravg (g (t)) :S -A (g (t)),